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Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\Lambda(=\mathbb{F}^{n^3})\) where \(\mathbb{F}\) is a field with \(|\mathbb{F}|>2\) be the space of structure vectors of algebras having the \(n\)-dimensional \(\mathbb{F}\)-space \(V\) as the underlying vector space. Also let \(G=GL(V)\). Regarding \(\Lambda\) as a \(G\)-module via the `change of basis' action of \(G\) on \(V\), we determine the composition factors of various \(G\)-submodules of \(\Lambda\) which correspond to certain important families of algebras. This is achieved by introducing the notion of linear degeneration which allows us to obtain analogues over \(\mathbb{F}\) of certain known results on degenerations of algebras. As a result, the \(GL (V)\)-structure of \(\Lambda\) is determined. degeneration; algebra; trace form; module; general linear group Fibrations, degenerations in algebraic geometry, Group actions on affine varieties, Representation theory of groups Linear degenerations of algebras and certain representations of the general linear group

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Taking as a model the completed theory of vector space endomorphisms, the present text aims at extending this theory to endomorphisms of finitely generated projective modules over a general commutative ring; now analogous results often require totally different methods of proof. The first important result is a structure theorem for such modules when the characteristic polynomial of the endomorphism is separable. The second topic deals with the minimal polynomial, whose mere existence is shown to require additional hypotheses, even over a domain. In the third topic we extend the classical notion of 'cyclic modules' as the modules which are invertible over the ring of polynomials modulo the characteristic polynomial. { } Regarding the diagonalization of endomorphisms, we show that a classical criterion of being diagonalizable over some extension of the base field can be transferred nearly verbatim to rings, provided that diagonalization is expected only after some faithfully flat base change. Many results that hold over a field, like the fact that commuting diagonalizable endomorphisms are simultaneously diagonalizable, hold over arbitrary rings, with this extended meaning of diagonalization. The Jordan-Chevalley-Dunford decomposition, shown as a particular case of the lifting property of étale algebras, also holds over rings.{ } Finally, in several reasonable situations, the eigenspace associated with any root of the characteristic polynomial is shown to be given a more concrete description as the image of a map. In these situations the classical theory generalizes to rings. projective modules; characteristic polynomials; eigenspaces; étale algebras; Jordan decomposition Projective and free modules and ideals in commutative rings, Algebraic systems of matrices, Relevant commutative algebra, Local structure of morphisms in algebraic geometry: étale, flat, etc., Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Linear transformations, semilinear transformations, Eigenvalues, singular values, and eigenvectors, Canonical forms, reductions, classification On the structure of endomorphisms of projective modules

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author extends standard monomial theory to the wonderful compactification \(X\) of a semisimple group \(G\) of adjoint type. Recall that \textit{R. Chirivì} and \textit{A. Maffei} have already constructed standard monomials for the more general situation of a wonderful compactification of a symmetric space [J. Algebra 261, No. 2, 310-326 (2003; Zbl 1055.14052)]. We fix a dominant weight \(\lambda\) and the corresponding line bundle \(\mathcal L_\lambda\) on \(X\). Then Chirivì and Maffei provide a basis of \(H^0(X,\mathcal L_\lambda)\) consisting of `standard monomials' with certain properties. The author shows that in the present situation one can do more. First of all, the standard monomials are shown to behave well with respect to restriction to \(B\times B\)-orbit closures, not just \(G\times G\)-orbit closures. Recall that there are finitely many \(B\times B\)-orbits and that they have been classified by \textit{T. A. Springer} [J. Algebra 258, No. 1, 71-111 (2002; Zbl 1110.14047)]. There are degrees of freedom in the construction of Chirivì, Maffei and these the author exploits to arrange more properties familiar from the classical standard monomial theory for flag varieties. The basis of \(H^0(X,\mathcal L_\lambda)\) is indexed by LS-paths again. And if \(Z\) is a \(G\times G\)-orbit closure, or more generally a \(B\times B\)-orbit closure, then the standard monomials that do not vanish on \(Z\) form a basis of \(H^0(Z,\mathcal L_\lambda)\). They can be characterized combinatorially. The many ingredients that are needed in the proof are explained clearly. standard monomials; group compactifications; wonderful compactifications; semisimple groups of adjoint type; orbit closures Appel, K, Standard monomials for wonderful group compactifications, J. Algebra, 310, 70-87, (2007) Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Standard monomials for wonderful group compactifications.

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. In the exact setting we can compute a \textit{standard basis} (the local equivalent of a Gröbner basis) using variations of Buchberger's algorithm. However in many practical situations this is not possible anymore, and we are forced to sort to approximate numerical data: examples of these situations are large systems of polynomials, where in practice the use of numerical algorithms is needed; if we want to examine then the ideal in the local ring at some solution point, this point is only known approximately, and as a result we need to work on an approximate setting, where usual algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it. Algorithms for computing the dual space of an ideal in a local ring, truncated at some degree, are known. These algorithms reduce the problem to one of linear algebra by using the singular value decomposition, which is a stable method. When the ideal is zero-dimensional, these truncated dual space algorithms provide a way to fully characterize the local properties of the ideal; also, the strategy can be adapted when the dimension is not zero, but it is known a priori. However, in any other case existing algorithms are not enough. The paper extends the strategy to the positive-dimensional case, and presents a stopping criterion based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions, which is discussed in the paper. Further applications involve developing numerical algorithms for computing the primary decomposition of an ideal. Numerical algebraic geometry; computational algebraic geometry; Hilbert function; Macaulay dual space; Macaulay2 Krone, Robert: Numerical algorithms for dual bases of positive-dimensional ideals. J. algebra appl. 12, No. 06, 1350018 (2013) Computational aspects in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symbolic computation and algebraic computation Numerical algorithms for dual bases of positive-dimensional ideals

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0619.00007.] This note gives an outline of the proof of the following. Let R be the local ring of a reduced plane curve singularity X. Then the torsion free rank 1 R-modules fall into finitely many 0-\ and 1-parameter families of isomorphism classes if and only if X is strictly unimodal (for contact equivalence), i.e. has modality \(\leq 2\) for right equivalence. - If X is not strictly unimodal, it deforms to at least one of \(J_{4,0}, X_{2, 0}, Z\) \(1_{2,0}\) or \(N_{16}\), and 2-parameter families can be explicitly constructed in these cases. The remaining cases are settled by direct calculation. First it is shown that a torsion free rank 1\ R-module admits a special type of generating system (called short systems), and then isomorphism is computed in the context of these systems. plane curve singularity; unimodal Schappert, A. : A characterization of strict unimodular plane curve singularities , in: Singularities, Representation of Algebras, and Vector Bundles, Lambrecht 1985 (Eds.: Greuel, G.-M.; Trautmann, G.). Lecture Notes in Math., Vol. 1273, Springer, Berlin- Heidelberg-New York (1987) pp. 168-177. Singularities of curves, local rings, Singularities in algebraic geometry, Multiplicity theory and related topics A characterisation of strictly unimodular plane curve singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A rank \(n\) local system \({\mathcal F}\) on \(\mathbb{P}^1_{\mathbb{C}} \backslash \{m\) points\} is a sheaf of complex vector spaces which is locally isomorphic to the constant sheaf \(\mathbb{C}^n\). There is a 1-1 correspondence between (equivalence classes of) rank \(n\) local systems and (equivalence classes of) \(n\times n\) first order linear differential equations on \(\mathbb{P}^1_\mathbb{C}\) having regular singularities at the \(m\) points. An irreducible local system is called rigid if the sheaf \(\text{End} ({\mathcal F})\) of the endomorphism of \({\mathcal F}\), extended to a sheaf on \(\mathbb{P}^1_\mathbb{C}\), has a trivial first cohomology group. Under the assumption that \({\mathcal F}\) is irreducible one can easily calculate the dimension of this first cohomology group from the local data, i.e., the conjugacy classes of the local monodromies around the \(m\) missing points. The dimension turns out to be \(2\rho\). In the general situation one can associate to \({\mathcal F}\) also an \(n\)-th order differential equation which has \(\rho\) as the number of ``accessory parameters''. Thus, roughly speaking, \({\mathcal F}\) is rigid if and only if the associated \(n\)-th order equation has no accessory parameters. Known examples of rigid differential equations are: ordinary hypergeometric equations, generalized hypergeometric equations, Pochhammer equations (i.e., the irreducible ones). This clearly shows that the theme of this book is a very classical one. The basic problems are: Irreducible recognition problem: Given the local data, determine whether they are attached to an irreducible local system. Irreducible construction problem: Suppose that the local data of an irreducible local system are given. Construct one or all irreducible local systems with these data. In the book these problems are solved under the condition that all data and local systems are rigid. The methods are hardly classical. Two operations on (rigid) local systems are introduced: middle convolution and middle tensor product. One makes use of: étale cohomology, perverse sheaves, Laumon's work on \(l\)-adic Fourier transforms, Deligne's proof of the Weil conjectures (and maybe more) to build the theory. The solution to the two problems turns out to be an algorithm, which starts with rank 1 local systems and uses the two operations above to obtain a rigid local system with the required data. If this algorithm does not produce a suitable rigid local system in a finite number of steps, then there is none. The last chapter of the book proves that a rigid local system is, in a certain sense, related to the Gauss-Manin differential equation on the De Rham cohomology of a family of algebraic varieties. The author's earlier proof of Grothendieck's conjecture for Gauss-Manin differential equations is extended to rigid local systems. Maybe accidently, the only rigid local systems, given explicitly in the book, are the three families mentioned above. For these families other proofs of Grothendieck's conjecture were already known. It is clear that this book presents highly important new views and results on the classical theory of complex linear differential equations. Unfortunately, it is equally clear that the researchers in the field of complex linear differential equations will recognize nothing in the book (except for the hypergeometric equation on page 7). rigid differential equations; hypergeometric equation; Pochhammer equations; rigid local system; Gauss-Manin differential equation; De Rham cohomology N.M. Katz, \(Rigid Local Systems\). Annals of Mathematics Studies, vol. 139 (Princeton University Press, Princeton, 1996) Local ground fields in algebraic geometry, \(p\)-adic differential equations, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Nonlinear differential equations in abstract spaces Rigid local systems

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a complex projective scheme and \(G\) a complex reductive group acting linearly on \(X\) with respect to an ample line bundle. According to Mumford's Geometric Invariant Theory (GIT), there is a projective scheme \(X/\!/G\) which is a categorical quotient of the open subscheme \(X^{\mathrm{ss}}\) of \(X\) consisting of points which are semistable for the given linearised action. The points of \(X/\!/G\) correspond to S-equivalence classes of semistable points. In [Mathematical Notes, 31, Princeton, New Jersey: Princeton University Press (1984); Zbl 0553.14020], the second author defined a stratification \(\{S_\beta:\beta\in{\mathcal B}\}\) of \(S\) into disjoint \(G\)-invariant locally closed subschemes, one of which is \(X^{\mathrm{ss}}\). This stratification depends on the lineaarisation and a choice of invariant inner product on the Lie algebra of a maximal compact subgroup of \(G\). It can be defined also using the results of \textit{G. Kempf} and \textit{L. Ness} [Lect. Notes in Math. 732, 233--243 (1979; Zbl 0407.22012)]. In this paper, the authors consider the problem of constructing a quotient for each unstable stratum \(S_\beta\). One can define a linearisation for the action of \(G\) on a projective completion \(\hat{S}_\beta\) of \(S_\beta\) and this canonical linearisation provides a categorical quotient for \(S_\beta\). However, the linearisation is not ample and the quotient collapses more orbits than one would like. To resolve this problem, the authors consider small perturbations of the canonical linearisation. This idea is then applied to the construction of moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data. Suppose that \({\mathcal F}\) has Harder-Narasimhan filtration \(0\subset{\mathcal F}^{(1)}\subset\cdots\subset{\mathcal F}^{(s)}={\mathcal F}\) and let \({\mathcal F}_i:={\mathcal F}^{(i)}/{\mathcal F}^{(i-1)}\). The Harder-Narasimhan type of \({\mathcal F}\) is the \(n\)-tuple \((P_1,\dots,P_s)\), where \(P_i\) is the Hilbert polynomial of \({\mathcal F}_i\). A sheaf \({\mathcal F}\) is said to be \(\tau\)-compatible if it possesses a filtration as above, where now the possibility that \({\mathcal F}^{(i)}={\mathcal F}^{(i-1)}\) is allowed. An \(n\)-rigidification for \({\mathcal F}\) is then an isomorphism \(H^0({\mathcal F}(n))\cong\oplus H^0({\mathcal F}_i(n))\) with the obvious compatibility relations. A concept of \(\theta\)-semistability for suitable \(\theta\in{\mathbb Q}^s\) can be defined and the following theorem (Theorem 8.9) proved. Let \(W\) be a projective scheme over \(\mathbb{C}\) and \(\tau\) a fixed Harder-Narasimhan type. For \(\theta\in{\mathbb Q}^s\) and \(n\gg0\), there is a projective scheme \(M^{\theta-ss}(W,\tau,n)\) which corepresents the moduli functor of \(\theta\)-semistable \(n\)-rigidified sheaves of Harder-Narasimhan type \(\tau\) over \(W\). The points of \(M^{\theta-ss}(W,\tau,n)\) correspond to S-equivalence classes of \(\theta\)-semistable \(n\)-rigidified sheaves with Harder-Narasimhan type \(\tau\). The construction of the stratification \(\{S_\beta\}\) for \(X\) a smooth projective variety is recalled in section 2, followed by the construction of quotients in section 3 and the extension to an arbitrary projective scheme in section 4. Simpson's construction of the moduli of semistable sheaves is recalled in section 5. In section 6, a stratification by Harder-Narasimhan types is described. The concept of an \(n\)-rigidified sheaf is introduced in section 7, followed by the construction of the moduli spaces in section 8. geometric invariant theory; sheaves; Harder-Narsimhan filtrations; rigidification; moduli spaces Hoskins, V; Kirwan, F, Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type, Proc. Lond. Math. Soc., 105, 852-890, (2012) Geometric invariant theory, Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Momentum maps; symplectic reduction, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This book is a classical reference on resolution of algebraic surfaces, embedded into a non-singular projective 3-fold over a perfect field \(k\). It gives a self-contained exposition on results of Zariski and leads up to the author's extensions to the case of characteristic \(p>0\). The principal results are global resolution, global principalization, dominance, birational invariance of dimension of homology groups, as well as (for \(p\neq 2,3,5)\) uniformization and birational resolution. -- The first edition of this book appeared in 1966 (see Zbl 0147.20504). The second edition under review comes in a time of newly increasing interest in resolution of singularities [cf. the work of \textit{A. J. de Jong}, Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996) with an alternative approach, called ``alterations'']. For the recent generation of mathematicians, there may arise some difficulties with the terminology, like the direction of arrows in the introduction (the author is ``blowing down'' where others are ``blowing up''). Here are some remarks on the new edition: It contains an appendix on analytic desingularization in characteristic zero, where a recent short proof is presented. -- From the author's abstract: ``It is hoped that this will remove the fear of desingularization from young minds and embolden them to study it further.'' The proof ``was inspired by discussion with the control theorist Hector Sussmann, the subanalytic geometer Adam Parusiński, and the algebraic geometer Wolfgang Seiler. Once again this illustrates the fundamental unity of all mathematics \dots By an inductive procedure incorporating the principalization lemma, the hypersurface \(f\) is approximated by a binomial hypersurface, i.e. a hypersurface of the form \(X_1^e+ X_2^{b_{e2}}\cdot\dots\cdot X_n^{b_{en}}\), where \(e\) is a positive integer and \(b_{e2},\dots, b_{en}\) are nonnegative integers. The reduction lemma enables us to further arrange matters so that \(\overline{b}_{e2}+\dots+ \overline{b}_{en}<e\), where \(\overline{b}_{e2},\dots, \overline{b}_{en}\) are the residues of \(b_{e2},\dots, b_{en}\) modulo \(e\). This then is a prototype of a good point of a surface.'' The final step is the reduction of multiplicity for such points using a ``good point lemma''. -- The book concludes with an extended bibliography, including also \textit{H. Hauser}'s recent article on ``Seventeen obstacles for resolution of singularities'' [in: Singularities. The Brieskorn anniversary volume, Prog. Math. 162, 289-313 (1998)] which the reader may wish to consult for further developments. resolution of algebraic surfaces; resolution of singularities; analytic desingularization S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monogr. Math., Springer, Berlin 1998. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rational and birational maps, Modifications; resolution of singularities (complex-analytic aspects) Resolution of singularities of embedded algebraic surfaces.

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this paper we study the Grassmannian of submodules of a given dimension inside the radical of a finitely generated projective module \(P\) for a finite dimensional algebra \(\Lambda\) over an algebraically closed field. The orbit of such a submodule \(C\) under the action of \(\operatorname{Aut}_{\Lambda}(P)\) on the Grassmannian encodes information on the top-stable degenerations of \(P / C\). The goal of this article is to begin the study of the global geometry of the closures of such orbits. In dimension one, this geometry is determined by the local rings of singular points. The smallest dimension for which the global geometry is not determined by local data is two, and this case is our main focus. We give several examples to illustrate the interplay between the geometry of the projective surfaces which arise and the corresponding posets of top-stable degenerations. Grassmannian; local rings; projective surfaces Representations of associative Artinian rings, Rational and ruled surfaces, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Orbit closures and rational surfaces

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper is a proof in a special case of: Abhyankar's local factorization conjecture. Suppose \(R \rightarrow S\) is a birational extension of regular local rings of dimension \(n>2\) and \(V\) is a valuation ring of \(Q(S)\) such that \(V\) dominates \(R\). The conjecture is that there exists a regular local ring \(T\subset V\) such that \(V\) dominates \(T\) and \(T\) dominates both \(R\) and \(S\) and where \(R \rightarrow T\) and \(S \rightarrow T\) are products of monoidal transforms. This conjecture has been already proven in dimension~3 by \textit{S. D. Cutkosky} [Adv. Math. 132, No. 2, 167--315 (1997; Zbl 0934.14006)]. In this new paper, \(R\subset S\) are regular local rings essentially of finite type over a field \(k\) of dimension \(n>2\) of characteristic~0, Abhyankar's conjecture is proven in the case where the valuation \(v\) has rank~1 and maximal rational rank: the valuation group \(VQ(S)\subset \mathbb R\) and has rational rank \(n\). By a Cutkosky's theorem, we may suppose that \(R\subset S\) is monomial, i.e. there exist a regular system of parameters \((x_1,\dots,x_n)\) (resp. \((y_1,\dots,y_n)\)) of \(R\) (resp. \(S\)) such that \[ x_i=\prod_{1\leq j \leq n} y_j^{a_{ij}}. \] Let us define \(\overrightarrow{v}:=(v(x_1),\dots,v(x_n))^t\), \(\overrightarrow{w}:=(v(y_1),\dots,v(y_n))^t\), \(A:=(a_{ij})\): \(A\) is a \(n\times n\) matrix with coefficients in \(\mathbb N\), \(\overrightarrow{v}\) and \(\overrightarrow{w}\) are real vectors with strictly positive rationally independant coefficients. Then, it appears that, if you blow up \(S\) along \((y_1,y_2)\), you make ``permissible column addition'' on \(A\) and \(\overrightarrow{w}\): you just add the second column to the first if \(v(y_2)>v(y_1)\), and in \(\overrightarrow{w}\), you change \(w_2\)in \(w_2-w_1\). If you you blow up \(R\) along \((x_1,x_2)\) and that \(S(1)\) the localization of this b.u. at the center of \(v\) is dominated by \(S\), then you make to \(A\) and \(\overrightarrow{v}\) ``permissible row substractions''. The authors prove that by a sequence of ``permissible column additions'', row interchanges and ``permissible row substractions'' you can make \(A=\text{Id}\). When you translate in the language of blowing ups, it means that you reach \(T\) from \(S\) and \(R\) by very special monoidal transforms. The authors do not pretend to originality in the result: there is already a proof of \textit{K. Karu} [J. Algebr. Geom. 14, No. 1, 165--175 (2005; Zbl 1077.14017)]. But their proof (inspired by Christensen) is the most elementary I ever read in this topic and is understandable by any undergraduate student in mathematics. The reviewer has the feeling that, with the same techniques, it is possible to solve the following problem. Problem. Given \(R\subset S\) monomial, can you find a CNS condition on \((A,\overrightarrow{v},\overrightarrow{w})\) such that \(R\subset S\) is a product of monoidal transforms? regular; moniomalization Global theory and resolution of singularities (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry Factorizations of birational extensions of local rings

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be an algebraically closed field of characteristic \(p>0\) and \(X_1, X_2\) be two smooth proper connected curves. Let \(\sigma_i: X_i\to X_i\) be an automorphism of order \(p\) and denote by \(\sigma\) the automorphism \(\sigma_1\times \sigma_2: X_1\times X_2\to X_1\times X_2=:Y\). It is proved that the graph of the resolution of any singularity of \(Y/\langle\sigma \rangle\) is a star-shaped graph with three terminal chains when \(X_2\) is an ordinary curve of positive genus. The intersection matrix of the resolution has determinant \(\pm p^2\). The singularity is rational. It is proved that for any \(s>0\) not divisible by \(p\) there are resolution graphs of wild \(\mathbb{Z}/p\mathbb{Z}\) quotient singularities with one node, \(s+2\) terminal chains and intersection matrix having determinant \(\pm p^{s+1}\). product of curves; cyclic quotient singularity; rational singularity; wild; intersection matrix; resolution graph; fundamental cycle Lorenzini, D.: Wild quotients of products of curves (2012, preprint) Singularities in algebraic geometry, Local ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Multiplicity theory and related topics, Singularities of surfaces or higher-dimensional varieties Wild quotients of products of curves

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper considers generalities for localization complexes for varieties. Examples of these complexes are given by the Gersten resolutions in various contexts, in particular in \(K\)-theory and in étale cohomology. The paper gives a general notion of coefficient systems for such complexes, the so-called cycle modules. There are the corresponding ``complexes of cycles with coefficients'' and their homology groups, the ``Chow groups with coefficients''. For these some general constructions are developed: proper push-forward, flat pull-back, spectral sequences for fibrations, homotopy invariance and intersection theory. If one specializes the material to the case of Milnor's \(K\)-theory as coefficient system, one obtains in particular an elementary development of intersections for the classical Chow groups. This treatment is somewhat different to former approaches. The main tool is still the deformation to the normal cone. The major different is that homotopy invariance is not established alone for the Chow groups, but for the ``cycle complex with coefficients in Milnor's \(K\)-theory''. This enables one to keep control in fibered situations. The proof of associativity of intersections is based on a doubled version of the deformation to the normal cone. Chow groups with coefficients; complexes of cycles; localization complexes for varieties; cycle modules; intersections; Chow groups M.~Rost, \emph{Chow groups with coefficients}, Doc. Math. 1 (1996), No. 16, 319--393. https://www.elibm.org/article/10000028; zbl 0864.14002; MR1418952 Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Higher symbols, Milnor \(K\)-theory Chow groups with coefficients

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((A, {\mathfrak m})\) be a local ring and \(I\) an ideal in \(A\) having maximal analytic spread. The authors define a multiplicity \(\mu (I,A)\) for \(I\) which coincides with the usual multiplicity in case \(I\) is \({\mathfrak m}\)-primary. If \(K\) is an algebraically closed field, the Stückrad-Vogel intersection number for a \(K\)-rational component of an intersection in a projective space over \(K\) can be interpreted as such a multiplicity. Denote by \(G\) the associated graded ring \(\text{gr}_ I (A)\) of \(A\) with respect to \(I\). Then \(\mu (I,A) : = \sum e (G/{\mathfrak p})\) length\((G_{\mathfrak p})\), where \({\mathfrak p}\) runs over all associated prime ideals of \(G/{\mathfrak m} G\) in \(G\) having dimension \(\dim (G/{\mathfrak m} G)\), and \(e (G/{\mathfrak p})\) is the (usual) multiplicity of the graded ring \(G/{\mathfrak p}\). \(\mu (I,A)\) can be computed by means of `super-reductions' for \(I\) which generalize superficial systems of parameters for \(I\) in case \(I\) is \({\mathfrak m}\)-primary. local ring; maximal analytic spread; multiplicity; Stückrad-Vogel intersection number; associated graded ring Achilles, Rüdiger; Manaresi, Mirella, Multiplicity for ideals of maximal analytic spread and intersection theory, J. Math. Kyoto Univ., 0023-608X, 33, 4, 1029-1046, (1993) Multiplicity theory and related topics, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Multiplicity for ideals of maximal analytic spread and intersection theory

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a finitely generated field. We construct an \(n\)-dimensional linear system \(\mathcal{L}\) of hypersurfaces of degree \(d\) in \(\mathbb{P}^n\) defined over \(K\) such that each member of \(\mathcal{L}\) defined over \(K\) is smooth, under the hypothesis that the characteristic \(p\) does not divide \(\gcd(d, n + 1)\) (in particular, there is no restriction when \(K\) has characteristic 0). Moreover, we exhibit a counterexample when \(p\) divides \(\gcd(d, n + 1)\). linear system; hypersurface; finite fields; smoothness Projective techniques in algebraic geometry, Hypersurfaces and algebraic geometry, Finite ground fields in algebraic geometry Linear families of smooth hypersurfaces over finitely generated fields

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \({\mathcal F}\) be a reduced foliation on the complex projective plane \(P^2\), given locally by a 1-form \(AdX + BdY + CdZ\) of degree \(q\). There is a natural rational map \(\Phi : P^2 \to (P^2)^\vee\) defined on \(P^2 \backslash \text{Sing} ({\mathcal F})\) that associates to each point \(Q \in P^2\) the point \(\Phi (Q) \in (P^2)^\vee\) corresponding to the line defined by \({\mathcal F}\) at \(Q\). The coefficients of the 1-form defining \({\mathcal F}\) locally also define a 2-dimensional linear system of curves \(\{\lambda A + \nu B + \rho C\}\) defining \(\Phi\); this linear system is called the net of polars of \({\mathcal F}\) and \(\Phi\) is called the polarity map of \({\mathcal F}\). The authors study the relationship set up by the map \(\Phi\). After resolving the points of indeterminacy of \(\Phi\) (i.e., the singularities of \({\mathcal F}\) or the base points of the net of polars), the authors give an explanation of the behavior of the integral curves of the foliation. The polarity map \(\Phi\) is then applied to give a generalization of Plücker's formula as a way of relating the global and local invariants of a curve in \(P^2\) with respect to \(\Phi\). Explicit formulas are then given for these invariants. polar varieties; Plücker's formula Equisingularity (topological and analytic), Foliations in differential topology; geometric theory, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Pencils, nets, webs in algebraic geometry, Singularities of holomorphic vector fields and foliations, Singularities in algebraic geometry Polarity with respect to a foliation

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The aim of this paper is to join the algebraic and the geometric information on \(A\), where \((A,m)\) is an excellent two-dimensional normal local ring, using the theory of the Hilbert functions and the theory of the resolution of singularities. Inspired by \textit{T. Okuma} [Ill. J. Math. 61, No. 3--4, 259--273 (2017; Zbl 1402.14044)], the authors investigate the integrally closed m-primary ideals of elliptic singularities and of strongly elliptic singularities. The authors present a result characterizes algebraically the strongly elliptic singularities. They show that there exist excellent two-dimensional normal local rings having no strongly elliptic ideals. Finally, the authors present necessary and sufficient conditions for the existence of strongly elliptic ideals in terms of the existence of certain cohomological cycles. When there exist, they present an effective geometric construction. Hilbert coefficients; elliptic ideals; elliptic singularities; reduction number; 2-dimensional normal domain Integral domains, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Elliptic surfaces, elliptic or Calabi-Yau fibrations Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0527.00002.] The paper deals with some properties of the 3-dimensional singularities with equations arising from Arnold's list, considered over an algebraically closed field of characteristic \(\neq 2,3\). These singularities are absolutely isolated, i.e. they have resolutions obtained by successively blowing up points, the ''canonical'' resolutions. These were considered first by P. J. Giblin in the topological context. - The article under review gives the algebraic description of the canonical resolution together with the intersections. Fundamental cycles (i.e. minimal negative embeddings of the exceptional loci) are computed. A method of calculating some cohomology groups on the nonreduced exceptional divisors is developed and applied to the normal bundles and the structural sheaves, thus getting vanishing theorems, applied to study the local moduli of the exceptional loci (embedded into the canonical resolution). For the \(A_ n\)-resolutions, the moduli space turns out to be smooth, and the dimension of its tangent space is the number of irreducible components isomorphic to the ruled surface \(F_ 2\). deformation; exceptional divisor; \(D_ n\); \(E_ n\); 3-dimensional singularities; vanishing theorems; local moduli of the exceptional loci; canonical resolution; \(A_ n\) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Some properties of the canonical resolutions of the 3-dimensional singularities \(A_ n\), \(D_ n\), \(E_ n\) over a field of characteristic \(\neq 2\)

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system \((R,\varphi)\) is defined to be a local algebraic dynamical system if \(R\) is a noetherian local ring with maximal ideal \(\mathfrak m\) and \(\varphi:R\rightarrow R\) is a local homomorphism such that \(\varphi(\mathfrak m)R\) is \(\mathfrak m\)-primary. Such a situation arises for the local ring at a fixed point under a finite self-morphism of an algebraic variety. In this paper, the authors introduce the following new invariant, called the local entropy, and explore many properties: \[ h_{\mathrm{loc}}(\varphi) = \lim_{n\rightarrow \infty} \frac 1n \log\left(\mathrm{length}_R(R/\varphi^n(\mathfrak m )R)\right). \] In Theorem 1, they show that (1) \(0\leq h_{\mathrm{loc}}(\varphi) < \infty\), (2) if \(R\) is of dimension \(d\) and of embedding dimension \(\delta\), then any \(\varphi\) which is contracting (i.e. \(\varphi^\delta(\mathfrak m) \subset \mathfrak m^2\)) satisfies \(\delta h_{\mathrm{loc}}(\varphi) \geq d \log 2\), (3) if \(R\) is of characteristic \(p>0\), then \(h_{\mathrm{loc}}(\mathrm{Frob}_R) = d\log p\). The local entropy shares many properties with topological entropy. For example, the local entropy of the \(m\)-th iterate of \(\varphi\) is \(m h_{\mathrm{loc}}(\varphi)\), and the local entropy on a finite union of closed \(\varphi\)-stable subspaces is the maximum of the local entropies on the subspaces. The authors also compare \(h_{\mathrm{loc}}(\varphi)\) with invariants of endomorphisms on germs of analytic functions, studied in [\textit{C. Favre} and \textit{M. Jonsson}, Ann. Sci. Éc. Norm. Supér. (4) 40, No. 2, 309--349 (2007; Zbl 1135.37018)]. The authors then study regular local rings. \textit{E. Kunz} [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] proved that \(R\) of characteristic \(p>0\) is regular \(\Longleftrightarrow \mathrm{Frob}_R\) is flat \(\Longleftrightarrow \mathrm{length}(R/(\mathrm{Frob}_R(\mathfrak m)R) )= p^d\). In Theorem 2, they generalize this to any \((R,\varphi)\): \(R\) is regular \(\Longrightarrow \varphi\) is flat \(\Longrightarrow \mathrm{length}(R/(\varphi(\mathfrak m)R) ) = e^{h_{\mathrm{loc}}(\varphi)}\), and all three become equivalent when \(\varphi\) is contracting. This inspires the authors to define Hilbert--Kunz multiplicity for any \((R,\varphi)\) regardless of characteristic, generalizing the case of \(\mathrm{Frob}_R\) treated in [\textit{P. Monsky}, Math. Ann. 263, 43-49 (1983; Zbl 0509.13023)]. For non-regular rings, it still remains open if this multiplicity even exists. Finally, in Theorem 3, the authors show that any local algebraic dynamical system can be lifted to that on an equicharacteristic complete regular local ring. This is inspired by \textit{N. Fakhruddin} [J. Ramanujan Math. Soc. 18, No. 2, 109--122 (2003; Zbl 1053.14025)] and \textit{A. Bhatnagar} and \textit{L. Szpiro} [J. Algebra 351, No. 1, 251--253 (2012; Zbl 1254.37066)], where a polarized self-morphism of a projective variety is extended to an ambient projective space. local algebraic dynamical systems; local entropy; endomorphism of finite length; Kunz regularity; Hilbert-Kunz multiplicity [32] Mahdi Majidi-Zolbanin, Nikita Miasnikov &aLucien Szpiro, &Entropy and flatness in local algebraic dynamics&#xPub. Mat.57 (2013) no. 2, p.~509-544, doi:10.5565/PUBLMAT_5721Article | &MR~31 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Arithmetic dynamics on general algebraic varieties, Local structure of morphisms in algebraic geometry: étale, flat, etc., Arithmetic and non-Archimedean dynamical systems Entropy and flatness in local algebraic dynamics

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R,\mathfrak{m})\) be a two dimensional regular local ring with infinite residue field. The theory of complete (i.e., integrally closed) ideals in this setup was initiated by Zariski, with the initial motivation of studying under an algebraic perspective the problem of curves passing through a fixed set of points, with given multiplicities. Since then, several authors have worked on these and related aspects of the theory: for example, \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, Vol. I, 203--231 (1988; Zbl 0693.13011)], \textit{S. D. Cutkosky} [Invent. Math. 98, No. 1, 59--74 (1989; Zbl 0715.13012)] and \textit{C. Huneke} [Publ., Math. Sci. Res. Inst. 15, 325--338 (1989; Zbl 0732.13007)], just to mention some. Let \(I\) be a complete simple residually rational \(\mathfrak{m}\)-primary ideal of \(R\). The main focus of this article is the study of the singular points of \(\Sigma = \text{Bl}_I(R)\), the blow-up of \(I\). The authors show that \(\Sigma\) has either one or two singular points, which in any case are rational singularities. The number of such singular points depends on whether the largest base point \(S_I\) of \(I\) is a free \(R\)-module or not. In addition, the authors compute the multiplicity of \(\Sigma\) at the singular point (respectively, points) in terms of certain non-simple ideal (respectively, ideals) adjacent to \(I\). blow-up; simple ideal; multiplicities Regular local rings, Singularities of surfaces or higher-dimensional varieties, Special surfaces The blow-up of a simple ideal

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let V be a Zariski closed subset of \({\mathbb{R}}^ n\) and S the sheaf of rings of (rational) regular functions on V. As usual, the authors call locally ringed spaces isomorphic to \(X=(V,S)\) an affine algebraic variety. If \(F={\mathbb{R}}, {\mathbb{C}}\), or \({\mathbb{H}}\), a continuous F-vector bundle B over X is said to admit an algebraic structure if there exists a projective T-module P, with \(T=R\otimes_{{\mathbb{R}}}F\), where R is the ring of global cross sections of S, such that B is isomorphic to the bundle arising from P in the usual way. The problem of characterizing the bundles over X, admitting algebraic structures goes back about 20 years and is, as yet, unsolved for general X. In this paper four theorems are announced which yield the most extensive results on this problem obtained so far. These are too technical to be stated here and the reader is requested to consult the paper itself. The aim is to characterize algebraic bundles, for special X, cohomologically by means of Chern classes. algebraic structure of vector bundle; real algebraic geometry; affine algebraic variety; Chern classes DOI: 10.1090/S0273-0979-1987-15558-3 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Real algebraic and real-analytic geometry, Rational points Algebraic vector bundles over real algebraic varieties

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system \textit{H. Hironaka}'s monumental work [Ann. Math. (2) 79, 109-203, 205-326 (1964; Zbl 0122.38603)] gave a non-constructive, existence proof of resolution of singularities over fields of characteristic zero. Here we announce the following stronger form of resolution of singularities: Theorem. Let \(X\) be a reduced subscheme embedded in a scheme \(W\), smooth over a field \(k\) of characteristic zero, and let \({\mathcal I}(X)\) be the sheaf of ideals defining \(X\). There exists a proper, birational morphism \(\pi:W_r\to W\), obtained as a composition of monoidal transformations, such that if \(X_r\subset W_r\) denotes the strict transform of \(X\subset W\), then: (i) \(X_r\) is regular in \(W_r\) and \(\text{Reg} (X)\simeq \pi^{-1}(\text{Reg}(X))\) via \(\pi\). (ii) \(X_r\) has normal crossings with \(\pi^{-1} (\text{Sing} (X))\), which is a union of hypersurfaces with normal crossings. (iii) The total transform of \({\mathcal I}(X)\) at \({\mathcal O}_{W_r}\) factors as a product of an invertible sheaf of ideals \({\mathcal L}\) supported on the exceptional locus, times the sheaf of ideals defining the strict transform of \(X\) (i.e. \({\mathcal I}(X){\mathcal O}_{W_r}={\mathcal L}\cdot {\mathcal I}(X_r))\). Parts (i) and (ii) are the usual conditions of embedded desingularization (Hironaka's theorem). Part (iii) is new and provides, in an elementary way, equations defining the embedded desingularization, from the equations defining the original singular scheme \((X\subset W)\). This result answers a question formulated by A. Nobile, which was the starting point of this research. A complete proof can be found in a paper to appear. resolution of singularities; embedded desingularization Bravo A., Villamayor O.: Strengthening the theorem of embedded desingularization. Math. Res. Lett. 8, 79--89 (2001) Global theory and resolution of singularities (algebro-geometric aspects) Strengthening the theorem of embedded desingularization

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(k\) be an algebraically closed field of characteristic \(0\). The guiding problem of the paper is to find characterizations of the representation type in terms of properties of the associated geometric objects. In the paper in question, the geometric objects studied are rational invariants. More precisely, let \(\Lambda\) be a finite dimensional \(k\)-algebra and \(\mathbf d\) a dimension vector. One defines a variety \(\text{mod}_\Lambda (\mathbf d)\), called the variety of \(\Lambda\)-modules with dimension vector \(\mathbf d\), in such a way that it possesses an action of an algebraic group \(\text{GL}(\mathbf d)\), which is natural in the sense that the orbits correspond to the isomorphism classes of the \(\Lambda\)-modules with dimension vector \(\mathbf d\). Given an irreducible component \(C\) one may study the field \(k(C)^{\mathbf d}\) of the rational invariants with respect to this action. An irreducible component is called indecomposable if it contains a dense subset of indecomposable modules. The main result of the paper states that if \(\Lambda\) is either hereditary or canonical (in the sense of Ringel), then \(\Lambda\) is tame if and only if for each indecomposable irreducible component \(C\) the field \(k(C)^{\mathbf d}\) is either \(k\) or \(k (t)\). In the case of hereditary algebras the author gives also two alternative characterizations of the tame representation type. Despite proving the above mentioned interesting results the author introduces a useful method allowing to compare the fields of rational invariants for different algebras. In particular, he uses this method in order to reduce the proof in the case of the canonical algebras to the Kronecker quiver. quivers; canonical algebras; hereditary algebras; rational invariants; representation types; varieties of modules; tame representation type Carroll, A., Chindris, C.: On the invariant theory of acyclic gentle algebras. To appear in Transactions of the American Mathematical Society. Preprint available at arXiv:1210.3579 [math.RT] (2012) Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Representations of associative Artinian rings, Group actions on affine varieties, Geometric invariant theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Geometric characterizations of the representation type of hereditary algebras and of canonical algebras.

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper generalizes, to the frame of Henselian local rings, the following well-known result of local analytic geometry: Let \(p\) be an isolated point of multiplicity \(r\) of a local complex analytic variety \(V\). Then, after deforming the coefficients of the equations defining \(V\), there is a small neighborhood of \(p\) containing exactly \(r\) points (counted with multiplicity) of the deformed variety. In fact, the generalization provided in this paper has several forms. The first one is exactly the mentioned one concerning zeros. There are also two other forms involving the so-called border basis of a finitely presented A-algebra. Let us mention that the proofs are purely constructive. local Bézout theorem; Henselian rings; roots continuity; stable computations; constructive algebra Henselian rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Solving polynomial systems; resultants, Effectivity, complexity and computational aspects of algebraic geometry, Other constructive mathematics Local Bézout theorem for Henselian rings

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this note all rings are assumed to be commutative Noetherian with identity. By \(H^ i_ m(R)\) we denote the i-th local cohomology module of a local ring \((R,m)\) with respect to the maximal ideal m. If \({\mathfrak x}\) is the sequence of elements \(x_ 1,...,x_ n\) of a ring R, \(K_*({\mathfrak x},R)\) denotes the Koszul complex of R with respect to \({\mathfrak x}.\) Canonical element conjecture: Let \((R,m)\) be a local ring of dimension \(n.\) Let \(F_*\) be a free resolution of \(R/m.\) Then the canonical element conjecture asserts that, for every system of parameters \({\mathfrak x}=x_ 1,...,x_ n\) for R and for every map of complexes \(\psi_*\) from \(K_*({\mathfrak x};R)\) to \(F_*\), \(\psi_ n(1)\neq 0\), or equivalently, the element \(\eta_ x\) of \(Tor^ R_ n(R/m,R/({\mathfrak x}))\) induced by \(\psi_ n(1)\otimes 1_{R/({\mathfrak x})}\in F_ n\otimes R/({\mathfrak x})\) is not zero. Theorem 1.1. Let \((R,m)\) be a 3-dimensional local ring of depth 2. Suppose \(H^ 2_ m(R)\) is a finite direct sum of cyclic R-modules. Then the canonical element conjecture holds for R. - Theorem 1.2. Let \((R,m)\) be a 3-dimensional local ring. Suppose that \(H^ 2_ m(R)\) is cyclic. Then the canonical element conjecture holds for R. We refer the reader to the paper by \textit{M. Hochster} [J. Algebra 84, 503-553 (1983; Zbl 0562.13012)] for a full discussion of the canonical element conjecture. We note that in this paper it was shown that this conjecture implies most of the homological conjectures which are known to follow from the existence of big Cohen-Macaulay modules. local cohomology module of a local ring; Koszul complex; Canonical element conjecture; system of parameters Huneke, C.; Koh, J.: Some dimension 3 cases of the canonical element conjecture. Proc. amer. Math. soc. 98, 394-398 (1986) Complexes, Local rings and semilocal rings, Local cohomology and algebraic geometry Some dimension 3 cases of the canonical element conjecture

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a smooth \(n\)-dimensional projective variety defined over an algebraically closed field of characteristic 0. In general \(X\) cannot be embedded linearly normal in \(\mathbb{P}^{2n+1}\). A general linear subspace \(\Lambda\subset\mathbb{P}^N\) of dimension \(N-2n-2\) can be obtained as the linear span of \(N-2n-1\) general points of \(\mathbb{P}^N\). Taking the linear span \(\Lambda\) of \(N-2n-1\) general points \(P_1,\dots,P_{N-2n-1}\) on \(X\), the projection of \(\mathbb{P}^N\) to \(\mathbb{P}^{2n+1}\) induces a morphism \(Y\to \mathbb{P}^{2n +1}\). Here \(\pi:Y\to X\) is the blowing-up of \(X\) at those points \(P_i\). Letting \(M=\pi^*(L)\otimes{\mathcal O}_Y(-\sum^{N-2n-1}_{i=1}E_i)\) then the morphism is associated to the complete linear system \(|M|\) (with \(E_i=\pi^{-1}(P_i))\). In case it is an embedding of \(Y\) in \(\mathbb{P}^{2n +1}\) then we find a linearly normal embedding of \(Y\) in \(\mathbb{P}^{2n+ 1}\). However there are examples showing that \(|M|\) is not always very ample (e.g. in case \(X\) is a variety ruled by linear subspaces in \(\mathbb{P}^N)\). On the other hand in [\textit{M. Coppens}, Pac. J. Math. 202, 313--327 (2002; Zbl 1059.14011)] it is proved that \(M\) is very ample in case \(L\) is of type \(kL'\) with \(k\geq 3n+1\) for some very ample invertible sheaf \(L'\) on \(X\). This result is the motivation behind the following conjecture studied in this paper. A subscheme \(Z\) of \(X\) of type \((m_1; m_2;\dots;m_t)\) for some integers \(m_i\geq 0\) is a 0-dimensional subscheme of \(X\) having support at different points \(P_{i,j}\) with \(1\leq i\leq t\), \(1\leq j\leq m_i\) such that locally at \(P_{i,j}\) one has \(Z=iP_{i,j}\). A property is said to hold for a general subscheme \(Z\) of \(X\) of type \((m_1;m_2;\dots;m_t)\) if that property holds for a point \((P_{i,j})\in X^{\sum m_j}\) in a non-empty Zariski-open subset of \(X^{\sum m_j}\). Conjecture. Let \(X\) be a smooth \(n\)-dimensional projective variety and let \(L\) be a very ample invertible sheaf on \(X\). For each integer \(t>0\) there exists an integer \(a_0(t)\) (depending on \(X\) and \(L)\) such that the following holds for a general subscheme \(Z\) of \(X\) of type \((m_1; \dots;m_t)\). Let \(\pi:Y\to X\) be the blowing-up of \(X\) at the points \(P_{i,j}\) in \(\text{Supp}(Z)\), let \(E_{i,j}=\pi^{-1} (P_{i,j})\) and let \(a\geq a_0(t)\) be an integer. If \[ h^0(X;aL)-\sum^1_{i=1} m_i \cdot {n+i-1\choose n}\geq 2n+2 \] then \[ M=\pi^*(aL)\otimes{\mathcal O}_Y\left( -\sum^t_{i=1}i\cdot\left(\sum^{m_i}_{j=1}E_{i,j}\right)\right) \] is very ample on \(Y\). In this paper we are going to prove the conjecture under the additional assumption that there are enough simple points, i.e. under an assumption \(m_1\geq b(a)\). This lower bound on \(m_1\) is negligible with respect to \((L^n/n!)a^n\), the highest order degree term of the polynomial \(p_L(a)\approx\chi(aL)\) for large \(a\). Theorem. There exists an integer \(a_1(t)\) and a polynomial \(p_1(a)\) of degree \(n\) with highest degree term \((L^n/n!)\cdot a^n\) (both \(a_1(t)\) and \(p_1(a)\) depending on \(X\) and \(L)\) such that the following property holds for a general subscheme \(Z\) of \(X\) of type \((m_1;\dots;m_t)\). Let \(a\geq a_1(t)\) be an integer. If \[ \sum^t_{i=2}m_i\cdot{n+i-1\choose n}\leq p_1(a) \] and \[ h^0(X;aL)-\sum^t_{i=1}m_i\cdot{n+i-1\choose n}\geq 2n+2, \] then \[ M=\pi^*(aL)\otimes{\mathcal O}_Y\left(-\sum^t_{i=1}i \cdot\left( \sum^{m_i}_{j=1}E_{i,j}\right)\right) \] is very ample on \(Y\). very ample linear system; embeddings Embeddings in algebraic geometry, Projective techniques in algebraic geometry, Classical problems, Schubert calculus, Divisors, linear systems, invertible sheaves An asymptotic very ampleness theorem for blowings-up of smooth varieties at general points

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The main aim of this paper is to clarify the concept of complete intersection in local algebra. Two different definitions of complete intersection are in fact currently given. In this paper, the classical definition is referred to as the ``absolute'' definition, which says that the local ring \((R;\mathfrak{M})\) is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. The more usual definition, instead introduced by Scheja and popularized by Grothendieck, is mentioned here as the ``formal'' definition; it requires the same property for the completion \(\hat R\) of \(R\) with respect to the \(\mathfrak{M}\) topology. The authors show that the two definitions are equivalent if \((R,\mathfrak{M})\) is a one dimensional integral domain of arbitrary codimension; moreover, they produce an example of a dimension three integral domain which is ``formally'' but not ``absolutely'' complete intersection. They first prove a fundamental theorem, which, in short, says that if \(R\) contains a field of characteristic zero and and \( \pi: T\to \hat R\) is a surjective morphism defined on a complete regular local ring with embedding dimension equal to the embedding dimension of \(R\), then a presentation of \(R\) as an absolute complete intersection exists iff it can be obtained as a restriction of \(\pi\) to a convenient regular local subring \(S\). On this result are based their following arguments. In section one the authors prove that such an \(S\) exists provided \(R\) is a one dimensional integral domain. They summarize their procedure as follows. ``The method will be to inductively define subrings \(R_i\) of \(T/K^i\), where \(\hat R = T/K\), in such a way that the natural maps \(T/K^{i+1} \to T/K^i\) restrict to surjections \(R_{i+1}\to R_i\) and that \(T/K^i\) is naturally isomorphic to the completion of \(R_i\). In order to achieve this we must carefully lift elements from \(R_i\) to \(T/K^{i+1}\) to form \(R_{i+1}\), and this relies heavily on the fact that \(R\) is a one dimensional integral domain. We thus obtain an inverse system of rings \(\{R_i\}\) and the inverse limit \(S= \lim R_i\) we show has the required properties.'' Section 2 is devoted to the construction of a three dimensional local unique factorization domain, whose completion is \(T = [x,y,z,w]/ (x^2+y^2)\), which is not a quotient of a regular ring by an ideal generated by a regular sequence. Here is a description of the procedure, as given by the authors. ``We build an example with the property that if \(R\) can be lifted to a regular local ring \(S\) contained in \(R[[x,y,z,w]]\), than \(S\) must contain an element \(\Theta = f(1+z\omega + z^2h +xa +yb)\) where \(f= x^2 + y^2\) and \(\omega \in R\) is known, but we have no information about \(h,a,b\). We also equip \(R\) with a large collection of prime ideals whose extension to \(T\) contains \((x,y)T\) and the construction of each of these prime ideals guarantees the existence of an element in the lifting which is congruent to \(f\) modulo that particular prime ideal \(Q\). This new element and \(\Theta\) are unit multiples of each other and so their quotient will be in \(S\). So \(1+z\omega+z^2h \in R/Q\) and it follows that \(z(\omega+zh)\in R/Q\). We will construct \(R\) so that \(z\in R\) and so \(\omega +zh\) is in the quotient field of \(R/Q\). We can't control \(h\) but we do know that, for some fixed value of \(h\), \(\omega+zh\) must be in the quotient field of \(R/Q\) for every one of the primes we construct. So, for every possible value of \(h\), we construct a prime ideal \(Q_h\) such that \(\omega +zh\) is trascendental over \(R/Q_h\), and so certainly not in the quotient field. This contradiction rules out the possibility of a lifting.'' complete intersections completion Heitmann, R. C.; Jorgensen, D. A., Are complete intersections complete intersections?, J. Algebra, 371, 276-299, (2012), MR 2975397 Complete rings, completion, Linkage, complete intersections and determinantal ideals, Complete intersections Are complete intersections complete intersections?

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A cohomologically rigid (complex) local system is rigid. Over a smooth projective variety, \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003)] conjectured that rigid local systems should be motivic, and thus irreducible rigid local systems should be defined over a ring of integers. The authors prove a variant of this conjecture, Theorem 1.1 in the article, that says when the variety is quasiprojective, any irreducible cohomologically rigid complex local system with finite determinant and quasi-unipotent local monodromies at infinity is integral. This result is proved in the projective case in an earlier paper also written by the authors [``Rigid connections and $\mathbf{F}$-isocrystals'', Preprint , \url{arXiv:1707.00752}], which only invokes \textit{V. Drinfeld}'s existence theorem [Mosc. Math. J. 12, No. 3, 515--542 (2012; Zbl 1271.14028)] on companions, see [\textit{K. S. Kedlaya}, ``Étale and crystalline companions. I'', Preprint, \url{arXiv:1811.00204}] for the latest on companions. The strategy used in the projective case is summarized in the introduction. It relies on a counting argument on the isomorphism classes of cohomologically rigid local systems, which are finitely many. To generalize this to the quasiprojective case with quasi-unipotent monodromies at infinity, Prop. 2.3 in the article, which is attributed to Deligne, is crucial, as there are again finitely many isomorphism classes of cohomological rigid local systems, with the weights and monodromy at infinity being controlled. local systems; rigidity; integrality Variation of Hodge structures (algebro-geometric aspects), Finite ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Cohomologically rigid local systems and integrality

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors investigate a procedure to resolve singularities that has reasonable ``functorial'' properties. The main result is expressed in terms of ``marked ideals'', that is a 5-tuple \(\mathcal I = (M,N,E,I,d)\) where \(N\) is a subvariety of \(M\), both smooth over a field of characteristic zero, \(I\) is a coherent sheaf of ideals of \(N\), \(E\) is a normal crossings divisor of \(M\), transversal to \(N\) and \(d\) is a positive integer. The singular locus, or cosupport, of \(\mathcal I\) is the set of points \(x\) of \(N\) such that the order of the stalk \(I_x\) is \(\geq d\). One may define the transformation of such a marked ideal with center a suitable smooth subvariety of \(N\) (the \textit{admissible} transformations), the result is a new marked ideal. The objective is to obtain, by means of a finite sequence of admissible transformations, a marked ideal whose cosupport is empty. Such a sequence is called a resolution sequence. If this can be done in a reasonable constructive (or algorithmic) way other more classical desingularization theorems follow rather easily. (This is explained in the present article). The authors define a notion of equivalence of marked ideals as follows. A test transformation is either an admissible one, or one determined by the blowing-up of a center which is the intersection of two components of \(E\), or one induced by a projection \(M \times {\mathbb{A}}^1 \to M\). Two marked ideals \(\mathcal I = (M,N,E,I,d)\) and \(\mathcal J = (M,N',E,J,d')\) are equivalent if they have the same sequences of of test transformations. Then they prove that there is way to associate to each marked ideal a resolution sequence such that if \(\mathcal I\) and \(\mathcal J\) are equivalent, then the resolution sequence associated to \(\mathcal I\) is obtained by using the same centers as were needed for the sequence of \(\mathcal J\). Moreover, this procedure is compatible with smooth morphisms \(M' \to M\). This is what is meant by functoriality of the process. The procedure is a variant of that introduced by the authors in their fundamental paper \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)] incorporating some ideas from \textit{J. Włodarczyck}'s article [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. The exploitation of the explicit requirements on functoriality simplifies the verification of the fact that certain constructions are independent of the elections made, which is a hard problem in this type of desingularization work. The authors use the functorial character of the algorithm to show that it coincides with others recently introduced by J. Wlodarczyk and J. Kollár. resolution; blowing-up; marked ideal; admissible transformation; derivative ideals; test transformation Bierstone, E., Milman, P.: Functoriality in resolution of singularities. Publ. R.I.M.S. Kyoto Univ.~\textbf{44}, 609-639 (2008) Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Birational geometry, Modifications; resolution of singularities (complex-analytic aspects), Equisingularity (topological and analytic) Functoriality in resolution of singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathfrak K\) be a field, and let \(S=\mathfrak K[x_{ij_i}\mid 1\leq i\leq k,\, 1\leq j_i\leq m_i]\) be polynomial rings with \(n=:\sum_{i=1}^km_i\) variables. The rational normal \(\mathfrak K\)-scroll (abbreviated as a \(k\)-scroll) \(\mathcal S(m_1- 1, m_2 - 1, \ldots, m_k - 1)\) is the variety in \(\mathbb P^{n-1}\) defined by the ideal \(I_2(M)\), which is generated by \(2 \times 2\) minors of the \(2 \times (n - k)\) matrix \((A_1, A_2,\ldots, A_k)\), where \[ A_i=\begin{pmatrix}x_{i\,1}&x_{i\,2}&\ldots&x_{i\,m_i-1}\\ x_{i\,2}&x_{i\,3}&\ldots&x_{i\,m_i} \end{pmatrix}. \] When \(k=2\), a \(k\)-scroll is called a scroll. In Theorem \(2.2\), the authors observed that \(R=:S/I_2(M)\) is a Koszul ring. In section \(3\), the authors computed the Betti numbers of \(\mathfrak K\) over \(k\)-scrolls. When \(k=2\), the authors explicitly constructed a finite free resolution of \(\mathfrak K\) over \(R\) and in Theorem \(4.1\), they proved that it is a minimal FFR. Betti numbers; minimal free resolution; rational normal \(\mathfrak K\)-scrolls Syzygies, resolutions, complexes and commutative rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Linkage, complete intersections and determinantal ideals, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Rational and ruled surfaces Toward free resolutions over scrolls

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A power structure over a ring is a method to give sense to expressions of the form \((1+a_1t+a_2t^2+\cdots )^m\), where \(a_i\), \(i=1, 2,\ldots \), and \(m\) are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of \(\lambda \)-and power structures over some Grothendieck rings. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural \(\lambda \)-structures on it which lead to the same power structure. We show that this power structure is effective. In the terms of this power structure we write some equations containing classes of Hilbert-Chow morphisms. We describe some generalizations of this construction for maps of varieties with some additional structures. lambda-structure; power structure; complex quasi-projective varieties; maps; Grothendieck ring Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups (category-theoretic aspects), Varieties and morphisms, Finite groups of transformations in algebraic topology (including Smith theory) Power structure over the Grothendieck ring of maps

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system First recall some definitions. For an extension \(A\subseteq B\) of rings with unity, an element \(b\) of \(B\) such that \(h(b)=0\) and \(h'(b)\) is a unit of \(B\), for some polynomial \(h(X)\) in \(A[X]\), is called a Henselian element over \(A\). A local ring is regular if its maximal ideal is generated by \(d\) elements, where \(d\) is the maximal lenght of a chain of prime ideals. A valued function field \((F|K, v)\) is said to admit local uniformization if, for every finite subset \(Z\) of the valuation ring \({\mathcal O}_F\) of \(F\), there exists a subring \(R\) of \(F\), of finite type over \(K\), such that the field of fractions \(\text{frac}(R)\) of \(R\) is equal to \(F\), \(R_{{\mathfrak p}}\) is regular (where \(\{{\mathfrak p}= \{x\in R\mid v(x)> 0\}\)) and \(Z\subseteq R\). Let \(A\) be an integrally closed domain with quotient field \(L\), \(I(A)\) be the integral closure of \(A\) in the separable-algebraic closure \(L^{\text{sep}}\) of \(L\), and \({\mathfrak m}\) be a maximal ideal of \(I(A)\). The inertia field of \(L^{\text{sep}}|L\) with relation to \({\mathfrak m}\) is the subfield \(L^i\) of \(L^{\text{sep}}\) fixed by the group \(\{\sigma\in \text{Gal}(L^{\text{sep}}|L)\mid\forall x\in I(A): x- \sigma(x)\in F{\mathfrak m}\}\); we know that if \(A\) is the valuation ring of a valuation \(v\), then \(v(L^i)= v(L)\) and if \(L\subseteq F\subseteq L^i\) then the extension \((F|L, v)\) is said to be unramified. Now let \(F\subseteq L^i\) such that \(F|L\) is a finite extension, \(A^*= F\cap I(A)\) and \(B= A^*_{{\mathfrak m}\cap A^*}\). First the authors prove that there is \(\eta\in B\) such that \(F= L(\eta)\), the minimal polynomial \(h(\eta)\) of \(\eta\) over \(L\) lies in \(A[X]\) and \(\eta\) and \(h'(\eta)\) are units of B. Next they prove that \(B= A[\eta]_{{\mathfrak n}}\), where \({\mathfrak n}= A[\eta]\cap{\mathfrak m}\). In particular, for every finite \(Z\subseteq B\) there exists a unit \(u\) of \(B\) in \(A[\eta]\) such that \(Z\subseteq A[\eta,1/\mu]\). With the extra assumption that \(A\) is a valuation ring, for every \(b\in B\) there exist Henselian elements \(r\), \(s\) in \(B\) such that \(b\in A[\eta,r,s]\). They prove that there exists a finite valued field extension \((F|L,v)\), where \(F\subseteq L^i\), such that \({\mathcal O}_F\) is not a finitely generated \({\mathcal O}_L\) algebra. They give a list of equivalent conditions under which \({\mathcal O}_F\) is a finitely generated \({\mathcal O}_L\) algebra. For example, both of: there exists a unit \(u\) of \(B\) in \({\mathcal O}_L[\eta]\) such that \({\mathcal O}_F={\mathcal O}_L[\eta,1/u]\), and: there are Henselian elements \(r\), \(s\) in \({\mathcal O}_F\) such that \({\mathcal O}_F={\mathcal O}_L[\eta,r,s]\). The other conditions are related to \(\text{Spec}({\mathcal O}_L[\eta])\), and they hold if \(\text{Spec}({\mathcal O}_L)\) is well-ordered by inclusion. Now, let \((F|K, v)\) be a valued function field. If there exists a transcendence basis \(T\) of \(F|K\) such that \((K(T)|K,v)\) admits local uniformization, \(F\subseteq K(T)^i\) and \(v\) is trivial on \(K\), then \((F|K, v)\) admits local uniformization. In the appendix, they prove that if \(R\) is an integrally closed domain and \(R\subseteq S\) is an integral extension such that \(\text{frac}(S)|\text{frac}(R)\) is finite and \(\text{Spec}(R)\) is the union of finitely many chains, then \(\text{Spec}(S)\) is the union of finitely many chains. Henselian elements; locl uniformization; elimination of ramification Valuations and their generalizations for commutative rings, Valued fields, Valuation rings, Regular local rings, Singularities in algebraic geometry Henselian elements

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The main proposition, Theorem 1.2, is the existence for excellent Deligne-Mumford champ of characteristic zero of a resolution functor independent of the resolution process itself. Received wisdom was that this was impossible, but the counterexamples overlooked the possibility of using weighted blow ups. The fundamental local calculations take place in complete local rings, and are elementary in nature, while being self contained and wholly independent of Hironaka's methods and all derivatives thereof, i.e. existing technology. Nevertheless \textit{D. Abramovich} et al. [``Functorial embedded resolution via weighted blowing ups'', Preprint, \url{arXiv:1906.07106}], have varied existing technology to obtain even shorter proofs of all the main theorems in the pure dimensional geometric case. Excellent patching is more technical than varieties over a field, and so easier geometric arguments are pointed out when they exist. Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Very functorial, very fast, and very easy resolution of singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this survey article, the author explains several connections between the Betti numbers and shifts occurring in a graded minimal free resolution of the homogeneous coordinate ring of a finite set of points in \(\mathbb{P}^n\), and the geometry of the configuration of those points. The ranks of the modules forming the linear part of such a resolution can be computed using the Koszul complex and some easy linear algebra. The author reviews this method in a very elementary and readable style. She shows how it was used in a series of joint papers with \textit{M. E. Rossi} and \textit{G. Valla} to study the geometric meaning of the length of the linear part of that resolution. For instance, she sketches a new proof of the strong Castelnuovo lemma [cf. \textit{M. L. Green}, J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] based on this technique. Another case in which the Betti numbers and shifts in the minimal resolution have been studied extensively is the case of points in generic position. The author explains the minimal resolution conjecture (MRC) and the work that has been devoted to it. For \(n+2\leq s\leq n+4\) points in \(\mathbb{P}^n\), the precise geometrical conditions on the points under which MRC holds are described, and sketches of the proofs of those joint results with M. E. Rossi and G. Valla are given. The paper ends with an explanation of the Green-Lazarsfeld conjecture and an extensive list of references. For anyone interested in syzygies of sets of points in \(\mathbb{P}^n\), this survey can be whole-heartedly recommended as an excellent starting point. coordinate ring of a finite set of points; geometry of the configuration; minimal resolution conjecture Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Sets of points and their syzygies

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors consider local formally smooth morphisms \(u: A\to B\) of noetherian local rings with residue fields of characteristic \(p>0.\) Their main result: Suppose that (i) the residue field K of B has a separate p- basis over the residue field k of A (i.e. a p-basis \(x=(x_ i)_{i\in I}\) such that K:k(x) is a separable extension) or (ii) B is complete; then there exist a noetherian local A-algebra \(\tilde A\) and a local A- morphism \(\tilde u: \tilde A\to B\) such that (a) \(\tilde u\) is formally smooth again, (b) \(\tilde A\) has the same dimension as B, and (c) \(\tilde A\) has a ''good'' A-structure, more precisely: \(\tilde A\) is a localization of a polynomial ring over A in case (i), and at least an inductive limit of localizations of polynomial rings over A in case (ii). (This result is well-known and rather straightforward to prove if the extension K:k is separable.) The paper contains a detailed study of extensions K:k with a separate p-basis and also of extensions without such a basis. Furthermore the authors give an application to the Néron desingularization of arbitrary formally smooth morphism. formally smooth morphisms; Néron desingularization DOI: 10.1016/0021-8693(86)90087-6 Commutative ring extensions and related topics, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) A structure theorem on formally smooth morphisms in positive characteristic

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be a local field of characteristic \(0\) and residual characteristic \(p>0\). Let \(\mathbf B_{\text{dR}}^+\) be the ring containing the \(p\)-adic periods of the algebraic varieties defined over \(K\). This ring was defined by \textit{J. M. Fontaine} [Ann. Math. (2) 115, 529--577 (1982; Zbl 0544.14016); Fontaine, Jean-Marc (ed.), Périodes \(p\)-adiques. Astérisque 223, 59--111, Appendix 103--111 (1994; Zbl 0940.14012)]. It is proved that \(\mathbf B_{\text{dR}}^+\) is the completion of \(\bar{K}\), the algebraic closure of \(K\), with respect to a topology described intrinsically that involves semimultiplicative but not multiplicative norms on \(K\). This is given in Théorème 3.1. An explicit formula is found to describe an element of \(\bar{K}\) as an element of \(\mathbf B_{\text{dR}}^+\). This formula is not essential to prove that \(\bar{K}\) is dense in \(\mathbf B_{\text{dR}}^+\) but it is useful for visualizing the topology induced on \(K\) by \(\mathbf B_{\text{dR}}^+\). The first three sections of this paper are modifications of the appendix written by the author in [loc. cit]. Section 1 contains results on complete local algebras. Section 2 is dedicated to the formula for elements of \(\bar{K}\) as elements of \(\mathbf B_{\text{dR}}^+\). In Section 3 it is proved the density of \(\bar{K}\) in \(\mathbf B_{\text{dR}}^+\) and its consequences. In the last section, the results are extended to the case where the starting point is a Banach algebra, for instance, the Tate algebra \({\mathbb Q}_p\{T_1,\ldots,T_d\}\), instead of a local field. Local fields; \(p\)-adic periods; \(p\)-adic differentials Galois theory, Galois cohomology, Local ground fields in algebraic geometry A construction of \(\mathbf B_{\text{dR}}^+\)

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R,\mathfrak{m})\) be a Noetherian local ring of dimension \(d>0\). Let \(I_\bullet = \{I_n\}_{n\in \mathbb{N}}\) be a graded family of \(\mathfrak{m}\)-primary ideals of \(R\), that is, \(I_0 = R\) and \(I_nI_m \subseteq I_{n+m}\) for all \(n,m \in \mathbb{N}\). The study of the volume \[ \displaystyle \text{vol}(I_\bullet) = \limsup_n \frac{\ell_R(R/I_n)}{n^d/d!} \] of \(I_\bullet\) has attracted the attention of many researchers over the years. The most classical example of such an object is given by choosing \(I_n=I^n\) for all \(n\), in which case one obtains the Hilbert-Samuel multiplicity of an \(\mathfrak{m}\)-primary ideal \(I\). Starting with \textit{A. Okounkov} and his interpretation of asymptotic multiplicities as volumes of certain cones (the Okounkov bodies) [Invent. Math. 125, No. 3, 405--411 (1996; Zbl 0893.52004); Prog. Math. 213, 329--347 (2003; Zbl 1063.22024)], these methods led to an approach used by \textit{R. Lazarsfeld} and \textit{M. Mustaţă} [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 5, 783--835 (2009; Zbl 1182.14004)] and \textit{K. Kaveh} and \textit{A. G. Khovanskii} [Ann. Math. (2) 176, No. 2, 925--978 (2012; Zbl 1270.14022)], who studied graded linear systems on projective schemes. In a series of papers, \textit{S. D. Cutkosky} used different methods to study these objects and, among other achievements, his work shed light on existence of volumes as actual limits [Algebra Number Theory 7, No. 9, 2059--2083 (2013; Zbl 1315.13040)]. In [J. Algebra 442, 260--298 (2015; Zbl 1408.13063)], \textit{S. D. Cutkosky} showed that when \(R\) is a regular local ring and \(I_{n+1} \subseteq I_n\) for all \(n\) (that is, \(I_\bullet\) is a graded filtration), there exists a constant \(\gamma>0\), independent of \(n\), such that \(0 \leq \ell_R(R/I_{n+1})-\ell_R(R/I_n) < \gamma n^{d-1}\) for all \(n \geq 0\). In a sense, even if the function \(n \mapsto \ell_R(R/I_n)\) is not necessarily a polynomial for \(n \gg 0\), under these assumptions it still has some polynomial-like behavior. The main result contained in this article is a generalization of Cutkosky's result, where the assumptions that \(R\) is regular and that \(\{I_n\}\) is a graded filtration are removed. In this case, there exists \(\gamma>0\) such that \[ \ell_R(R/I_{n+1}) -\ell_R(R/I_n) < \gamma n^{d-1} \] for all \(n \geq 0\). Without the assumption that \(I_\bullet\) is a graded filtration, there is no hope of obtaining a finite lower bound. Furthermore, the constant \(\gamma\) is independent of \(n\), but the authors show in Section 5 that it may depend on the graded family \(I_\bullet\). The tools needed to prove the main theorem are developed in Sections 3 and 4. First, the authors analyze the case when the ideals of the graded family are generated by monomials in a polynomial ring over an Artinian ring. Then, they reduce the original problem to this case, by passing to the associated graded ring with respect to a parameter ideal. graded family; volume; multiplicity; asymptotic polynomial Multiplicity theory and related topics, Regular local rings, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves Growth of multiplicities of graded families of ideals

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author proves a series of properties for the Hodge structure of local systems, generalizing earlier results of himself and other authors. For example: Let \(X\) be a quasi-projective complex manifold and \(\rho:\pi_1(X)\rightarrow U(N,\mathbb C)\) a character inducing a rank \(N\) local unitary system \(V_{\rho}\) on \(X\). The cohomology groups \(H^i(X,V_{\rho})\) support a canonical mixed Hodge structure and a Hodge filtration \(\dots\subset F^{p+1}H^n(V_\rho)\subset F^{p}H^n(V_\rho)\subset\dots\). Consider the family \({\mathcal W}=\{V_\rho\otimes L_\chi\,|\,\chi\in \text{ Hom}(\pi_1(X),S^1)\,\}\), where \(L_\chi\) is the rank one unitary local system induced by \(\chi\). The author describes for \(l\in\mathbb N\) the structure of the family \(S^{n,p}_l=\{W\in {\mathcal W}\,|\,\text{dim}F^pH^n(W)/F^{p+1}H^n(W)\geq l\,\}\) under the additional assumption that \(H^1(\overline{X},\mathbb C^*)=0\) for some non-singular compactification \(\overline{X}\) of \(X\). Assuming that \(H_1(X,\mathbb Z)\) is torsionfree and identifying the universal cover of the group Hom \((\pi_1(X),S^1)\) with the tangent space \(T\) at the neutral element , the universal covering map with the exponential map and the fundamental domain \({\mathcal U}\) of the \(\text{ Hom }(\pi_1(X),S^1)\)-action on \(T\) with the unit cube in \(T\) he proves that the family \(S^{n,p}_l\) is a finite union of polytopes in \({\mathcal U}\). The proof is based on a study of Deligne extensions of local systems on \(X\) to bundles on \(\overline{X}\) [\textit{P. Deligne}, Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics. 163. (Berlin-Heidelberg-New York): Springer-Verlag. (1970; Zbl 0244.14004)]. Another main result of the article deals with a local version of this setting: Let \({\mathcal X}\) be a germ of a complex space with isolated normal singularity such that the intersection of \({\mathcal X}\) with a sufficiently small sphere around the singularity is simply connected , and let \({\mathcal D}\) be a divisor on \({\mathcal X}\) with \(r\) irreducible components and \(X:={\mathcal X}\backslash {\mathcal D}\). The family of rank one local systems is then parametrized by the affine torus \(H^1(X,\mathbb C^*)=\mathbb C^{*r}\). The author proves that the characteristic varieties \[ S^n_l=\{\chi\in H^1({\mathcal X}\backslash{\mathcal D},\mathbb C^*)\,|\,\text{ dim }{\mathcal X}\backslash{\mathcal D},L_\chi)\geq l\,\}, 1\leq n\leq\text{dim }X, \] of \(({\mathcal X},{\mathcal D})\) have a decomposition as a finite union of subtori of \(H^1({\mathcal X}\backslash{\mathcal D},\mathbb C^*)\), translated by points of finite order. For the proof he uses a Mayer-Vietoris spectral sequence for the union of tori bundles on quasi-projective manifolds which degenerates in the term \(E_2\), and also the results of \textit{D. Arapura} [J. Algebr. Geom. 6, No. 3, 563--597 (1997; Zbl 0923.14010)] in the global Kähler setting. The author also introduces twisted characteristic varieties as a multivariable generalization of twisted Alexander polynomials. He uses these varieties to obtain information about the homology of non unitary local systems. Finally explicit calculations of the graded components of the Hodge filtration are given for several examples where \(X\) is the complement of an arrangement of hypersurfaces in \(\mathbb P^n, n\leq 3\). unitary local system; Hodge filtration; Hodge number; polytope; Alexander polynomial; Deligne extension Libgober, A.: Alexander invariants of plane algebraic curves. Singularities, Part 2 (Arcata, Calif., 1981), pp. 135-143. Proceedings of Symposia in Pure Mathematics, vol. 40. Amer. Math. Soc., Providence (1983) Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), de Rham cohomology and algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Knots and links in the 3-sphere Non vanishing loci of Hodge numbers of local systems

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Generalizing a construction of [\textit{A. Girand}, Bull. Soc. Math. Fr. 144, No. 2, 339--368 (2016; Zbl 1344.14011)], the author introduces an \(n\)-parameter family \(\nabla_{\lambda}\) of flat, regular singular \(\mathfrak{sl}_{2}\)-connections on the trivial bundle of complex projective space \(\mathbb{P}^{n}\). Let \(D\) be the polar divisor of \(\nabla_{\lambda}\).\\ The monodromy representation of \(\nabla_{\lambda}\) is computed for generic values of \(\lambda\). The author describes the local monodromy explicitly for a specific choice of generators for the fundamental group of \(\mathbb{P}^{n}\setminus D\), see Proposition 3.3. This shows that the monodromy group lies inside the infinite dihedral group \(\mathbf{D}_{\infty}\). In addition the monodromy representation is proven to be virtually abelian, that is it is abelian after passing to a finite cover of \(\mathbb{P}^{n}\setminus D\), see Theorem 1.3. \\ Using \(\nabla_{\lambda}\) the author constructs via pullback an isomonodromic family of connections on \(\mathbb{P}^1 \times T\) where \(T\subset \mathbb{A}^{2(n-1)}\) is a certain Zariski open subset parametrizing generic lines in \(\mathbb{P}^{n}\). From this isomonodromic family the author obtains an algebraic solution to the Garnier system in \(2(n-1)\) variables, see Theorem 1.4. algebraic function; Garnier system; isomonodromic deformation Algebraic functions and function fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This interesting paper extends properties of pencils and other linear systems given by \textit{O. Zariski} [Trans. Am. Math. Soc. 50, 48-70 (1941; Zbl 0025.21502)] to any projective or affine variety by looking at linear families and systems defined in a much larger setting. For instance, let A be a commutative, noetherian, integral algebra over a field \(k\), define in \(W\), a vector space of dimension \(r+1\) (\(r>0)\), \(W\subset A\), an equivalence relation \(f\sim g\) if \(f=\alpha g\), \(\alpha \in k\). Then \(F=(W-\{0\})/\sim\) is a linear family of dimension \(r\) in \(A\). -- After a careful study of the resulting linear systems, irreducible systems are considered, similarities -- and differences -- with the classical case are noted, an extension of Bertini's theorem to non-normal varieties is proven and irreducibilities of specific linear families are studied. Bertini theorem; irreducibilities of linear families Divisors, linear systems, invertible sheaves About some algebraic aspects of linear systems and of the theorem of Bertini

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A complete characterization of the categorical quotients of \(X=(\mathbb{P}^1)^n\) by the diagonal action of \(G=\text{SL}(2,\mathbb{C})\) with respect to any polarization \(m=(m_1,\dots,m_n)\) is given by \textit{M. Polito} [C. R. Acad. Sci., Paris, Sér. I 321, 1577--1582 (1995; Zbl 0860.14042)]. In this paper under review, the categorical quotient \(X^{ss} (m)//G\) is obtained by a linear system on \(\mathbb{P}^{n-3}\) depending on the polarization. More precisely, for a given polarization \(m=(m_1,\dots,m_n)\) such that \(| m|=\sum m_i=2r+2\) (if \(| m|\) is odd, then replace \(m\) by \(2m)\), \(2m_i\leq| m|\) for all \(i\) and \(n\geq 3\), we consider the linear system \(\Lambda(m)\) of \((r-m_n+1)\)-forms on \(\mathbb{P}^{n-3}\) vanishing with multiplicity \(r-(m_n +m_j)+1\) at \(e_j\) \((1\leq j\leq n-1)\), where \(e_1,e_2,\dots,e_{n-1}\) are \(n-1\) points in \(\mathbb{P}^{n-3}\) in general position. The rational map \(\iota_{\Lambda(m)}:\mathbb{P}^{n-3}\to| \Lambda(m)|^*\) induced by the linear system \(\Lambda(m)\) maps \(\mathbb{P}^{n-3}\) onto \(X^{ss} (m)//G\). If the set of stable points \(X^s(m)\) is non-empty, then \(\iota_{ \Lambda(m)}\) is birational. As an application of this result, we deduce the following characterization: If \(n\geq 5\) and \(m_1\leq m_2\dots\leq m_n\leq r\), then \(X^{ss}(m)//G\simeq\mathbb{P}^{n-3}\) (or \(X^{ss} (m)// G\simeq (\mathbb{P}^1)^{n-3})\) if and only if \(m_n\leq r<m_n+m_1\) (respectively, \(m_n+m_1+ m_2+\cdots+m_{n-3} \leq r+1)\). categorical quotients of projective space; GIT quotients; group actions; linear systems; polarization; semistable points Geometric invariant theory, Group actions on varieties or schemes (quotients), Divisors, linear systems, invertible sheaves Linear systems and quotients of projective space

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors construct free resolutions for the coordinate rings of some subvarieties of Grassmannians \(G(n,N)\), over the complex field. Considering \(G(n,N)\) as the quotient of \(\mathrm{GL}_N\) by the action of a parabolic group \(P\), then its Schubert subvarieties \(X(w)\) correspond to orbits of (the classes of) elements \(w\in\mathrm{GL}_N\) under the action of the group of upper triangular matrices. Define the \textit{opposite big cell} \(O^-\) as the orbit of the identity under the group of lower triangular matrices. The authors consider \textit{opposite cell varieties}, defined as intersections \(Y(w)=X(w)\cap O^-\). The authors determine a free resolution for opposite cell varieties \(Y(w)\). The construction is effective when \(w\) is suitably chosen in \(\mathrm{GL}_N\). Opposite cell varieties for which the construction is effective fill a wide class of subvarieties of Grassmannians, which properly includes determinantal varieties. The method relies in the computation of the cohomology of a vector bundle over a desingularization of \(Y(w)\). The bundle is a restriction of a homogeneous bundle over a quotient \(P/\tilde P\), for a suitable subgroup \(\tilde P\). The bundle is not completely reducible, yet the authors are able to compute its cohomology, by restricting to the pieces of a suitable filtration of \(P/\tilde P\). Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Syzygies, resolutions, complexes and commutative rings, Determinantal varieties Free resolutions of some Schubert singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Constructive desingularization of varieties (in characteristic zero) is a very important tool in modern algebraic geometry, but unfortunately implementations for the general case (i.e. arbitrary dimension) suffer from the high computational complexity of the construction itself. In the case of curves and surfaces, other approaches were already known, when Hironaka proved his famous result on the existence of resolution of singularities in characteristic zero in the 1960s, but were no longer in the center of interest after the general case was proved. The important exception here is the curve case, where the Newton-Puiseux expansion locally provides the desired data and is successfully applied for many practical purposes. In this article, the author studies desingularization of hypersurfaces in \({\mathbb P}^3\) over a field of characteristic zero on the basis of these considerations, using Jung's (local) approach to resolution of surface singularities. To this end, he first discusses the formal setting in which he is working and develops Jung's construction to the point where he can give explicit (a priori theoretical) algorithms for its various subtasks. To turn these considerations into an implementation, he then passes to a discussion of the practical aspects of multivariate algebraic power series and gives a new proof of the fact that the concept of rational Puiseux parametrizations can be extended to multivariate quasiordinary polynomials. At the end of the article, the author also mentions the performance of his implementation of the described algorithm in MAGMA and its availability, as well as the still open problem to obtain a dual graph of the resolution along these lines. resolution of singularities; desingularisation of surfaces; algebraic power series; Jung's algorithm; Hirzebruch-Jung desingularisation; desingularization; surface Tobias Beck, Formal desingularization of surfaces --- The Jung method revisited, Tech. Report 2007-31, RICAM, December 2007 Global theory and resolution of singularities (algebro-geometric aspects) Formal desingularization of surfaces: The Jung method revisited

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system One of the results in this article may be considered more or less known to specialists, but apparently it has no rigorous proof in the existing literature: Theorem. The normal crossings locus \(X_{nc}\) of an algebraic variety \(X\) is open in \(X\). It should be mentioned, that the corresponding analytic case is not that difficult, but here \(X\) is an algebraic variety over an algebraically closed field \(k\) of any characteristic. Applications of the above theorem can be found in the resolution theory of singularities. It is a typical way of reasoning to transfer analytic to algebraic results relating the algebraic local ring of \(x\in X\) and its completion or Henselization (the ``intermediate object''). They are obtained via étale topology and the celebrated Artin approximation theorem. This paper shows, there may still be surprises in that direction. The properties studied are local, thus it is sufficient to consider subvarieties \(X\) of the affine \(n\)-space. \(X\) is said to be algebraically normal crossings (anc) at \(p\in X(k)\), if it is defined near \(p\) by an equation \(y_1 \cdot \dots \cdot y_l\) (\(l\leq n\)), where \(y_i\) are local coordinates (i.e. a regular system of parameters for the local ring of \(X\) at \(p\)). \(X\) is said to be normal crossings (nc) at \(p\), if the completion \(\hat{X}_p\) is anc at \(p\). First of all it is shnown, that \(X_{anc}\) is open when \(X\) is a finite union of hypersurfaces. Then -- more generally -- sets of points \(q\in X\) are discussed, for which \(\hat{\mathcal O}_{X,q}\) has some given property. Are these subsets of \(X\) Zariski-open (resp. closed or locally closed)? Again the appropriate techniques arise from étale und formal neighbourhoods, related as mentioned above. This leads to the theorem on openness of \(X_{nc}\); as a corollary, the openness of the monomial locus \(X_{mon}\) of a hypersurface is shown. The article is illustrated with several instructive examples: It is natural to ask for other local properties of schemes which give rise to results similar as in the cases before. This is discussed here for Mikado points (as considered by H. Hauser) and for formally irreducible points. The examples worked out in detail show that such expectations fail in general. étale neighbourhood; Artin approximation; normal crossings; normal crossing locus; monomial locus Étale and flat extensions; Henselization; Artin approximation, Formal neighborhoods in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Formal power series rings Étale neighbourhoods and the normal crossings locus

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S=\{t_{1},\ldots ,t_{p+1}\}\subset \mathbb C\mathbb P^{1}\), \(X=\mathbb C\mathbb P^{1}\backslash S\) and \({\mathcal A}=\{A_{1},\ldots ,A_{p+1}\}\subset \text{GL}(n,\mathbb C)\) satisfying \(A_{p+1} \cdots A_{1}=I_{n}\). Then associating elements from \({\mathcal A}\) to some \((p+1)\) distinct loops around points in \(S\), starting from a different point, one defines a local system of rank \(n\) over \(X\), thus a representation of \(\pi _{1}(X)\). First \textit{N. H. Katz} [Math. Res. Lett. 3, No. 4, 527--536 (1996; Zbl 0889.42014)] gave an algorithm for obtaining all rigid local systems, then \textit{M. Dettweiler} and \textit{S. Reiter} [J. Symb. Comput. 30, No. 6, 761--798 (2000; Zbl 1049.12005)] reformulated the algorithm using a generalization of the circuit matrices for the Pochhammer differential equation. Using the results of the second author [Math. Nachr. 279, No. 3, 327--348 (2006; Zbl 1112.34024)] and a result of \textit{V. Kostov} on the additive version of the Deligne-Simpson problem [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 8, 657--662 (1999; Zbl 0937.20024)], the authors obtain a new and different method for obtaining all rigid local systems of semi-simple type. They follow the construction of Fuchsian systems of differential equations with monodromy representations corresponding to rigid local systems of semi-simple type, which give an explicit solution of the Riemann-Hilbert problem for such representations. They also prove that such Fuchsian systems of differential equations have integral representations of the solution, i.e. every section of every rigid local system of semi-simple type has an integral representation. Fuchsian system; monodromy representation; Pochhammer differential equation; Deligne-Simpson problem; Riemann-Hilbert problem Haraoka Y., Yokoyama T.: Construction of rigid local systems and integral representations of their sections. Math. Nachr. 279, 255--271 (2006) Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain Construction of rigid local systems and integral representations of their sections

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The aim of this paper is to generalize to general schemes of finite type over a field a principalization algorithm for monomial schemes due to \textit{R. A. Goward jun.} [Trans. Am. Math. Soc. 357, No. 12, 4805--4812 (2005; Zbl 1079.14021)] in the non-singular setting. If \(Y_1,\dots, Y_n\) are Cartier divisors of a scheme \(X\) (of finite type over a field) we say that \(\{Y_1,\dots, Y_n\}\) has \textit{regular crossings} if for every \(A \subset \{Y_1,\dots, Y_n\}\) and every point \(p \in \bigcap_{Y \in A}Y\), the local equations for the \(Y\in A\) form a regular sequence at \(p\). Then the main theorem is that if \(D_1,\dots, D_h\) are linear combinations of the \(Y_i\) with positive coefficients, then the ideal sheaf \(\mathcal I_{D_1} + \cdots + \mathcal I_{D_h}\) can be transformed into a principal ideal sheaf by a sequence of monomial blowups. principalization; monomial scheme; blowups Harris, C.: Monomial principalization in the singular setting. J. commut. Algebra 7, 353-362 (2015) Singularities in algebraic geometry Monomial principalization in the singular setting

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(x\) be a singular point of a projective surface \(S\subset \mathbb{P}^r\) and let \((A,m)\) be the local ring of \(S\) at \(x\). Call \(G\) the corresponding tangent cone. The author assumes that the normalization \(\overline A \) of \(A\) is regular and \(G\) splits in a union of planes, in such a way that its projectification is a union of lines in generic position (i.e. with generic Hilbert function). With this setting, the author proves that the conductor of \(A\) in \(\overline A\) is a power of \(m\). This result extends to higher dimension the known fact that a curve singularity, whose projectivized tangent cone is a set of points in generic position, has conductor equal to a power of the maximal ideal. conductor; tangent cone; normalization Orecchia, F.: On the conductor of a surface at a point whose projectivized tangent cone is a generic union of lines. In: Proceedings of Lecture Notes in Pure and Applied Mathematics, vol. 217. Dekker, New York (1999) Regular local rings, Singularities of surfaces or higher-dimensional varieties On the conductor of a surface at a point whose projectivized tangent cone is a generic union of lines

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The main theme of this paper is a systematical study on characteristic variety of a holonomic system with regular singularity on a complex manifold X, especially extensions of index theorems on characteristic cycles. A holonomic system of \({\mathcal D}_ X\)-modules is originally introduced by Sato-Kawai-Kashiwara in 1972, which is a \({\mathcal D}_ X\)- module \({\mathcal M}\) whose dimension of its characteristic variety SS(\({\mathcal M})\) coincides with the dimension of X. For two holonomic systems \({\mathcal N}\) and \({\mathcal M}\) with regular singularity, the author of this paper proves a global index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})=I(SS({\mathcal M}),SS({\mathcal N}))\) where I(A,B) is the multiplicity of the intersection of the cycles A and B. Moreover the author proves the local index theorem: \(\Sigma_ k(- 1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})_ X=I_ X(SS({\mathcal M}),SS({\mathcal N})),\) and the microlocal index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal E}_ X^{{\mathbb{R}}}}({\mathcal N}^{{\mathbb{R}}},{\mathcal M}^{{\mathbb{R}}})_{\xi}=I_{\xi}(SS({\mathcal M}),SS({\mathcal N})).\) The present work arise from the author's interest to characteristic varieties of the highest-weight modules over complex semi-simple Lie algebras. The detailed treatment of that subject appears in the next paper [Adv. Math. 61, 1-48 (1986)]. index theorems on characteristic cycles; holonomic system of \({\mathcal D}_ X\)-modules; characteristic varieties Ginzburg, V., \textit{characteristic varieties and vanishing cycles}, Invent. Math., 84, 327-402, (1986) Complex manifolds, Analytic subsets and submanifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps Characteristic varieties and vanishing cycles

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Inclusions of regular local rings \(R\subset S\) of dimension two with common quotient field have (according to a well known theorem of Zariski-Abhyankar) a simple structure, namely: \(R\subset S\) can be factored by a unique finite product of quadratic transforms. In dimension \(\geq 3\) the situation is a lot more complicated. In this sense Abhyankar made the following conjecture [see \textit{S. S. Abhyankar}, ``Ramification theoretic methods in algebraic geometry'', Ann. Math. Stud. 43 (1959; Zbl 0101.38201)]: Assume that \(K\) is a field of algebraic functions over a field \(k\), and \(R\) and \(S\) are regular local rings, essentially of finite type over \(k\), with quotient field \(K\). Let \(V\) be a valuation ring which dominates \(R\) and \(S\). Then there exists a regular local ring \(T\), essentially of finite type over \(k\), with quotient field \(K\), dominated by \(V\), containing \(R\) and \(S\), such \(R\subset T\) and \(S\subset T\) can be factored by products of monoidal transforms. The aim of the present paper under review is to prove a fundamental local theorem that implies Abhyankar's conjecture in dimension \(3\). Using his result, the author also proves the following global result (which partially answers a question of Hironaka and Abhyankar): Let \(k\) be a field of characteristic zero, \(\varphi:X \to Y\) a birational morphism of integral nonsingular proper excellent \(k\)-schemes of dimension \(3\). Then there exists a nonsingular proper \(k\)-scheme \(Z\) and birational morphisms \(f:Z\to X\) and \(g:Z\to Y\) such that \(\varphi \circ f=g\), with \(f\) and \(g\) locally products of monoidal transforms. regular local rings; valuation ring; quadratic transforms; monoidal transforms; birational morphism Cutkosky S.D.: Local factorization of birational maps. Adv. Math. 132(2), 167--315 (1997) Rational and birational maps, Valuation rings, Extension theory of commutative rings Local factorization of birational maps

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper is in some sense a follow-up to the paper ``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'' by the first author et al. [``Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes'', Preprint, \url{arXiv:0905.2191}]. In that paper, it was proved that every reduced excellent Noetherian scheme \(X\) of dimension at most two admits a resolution of singularities via finitely many ``permissible'' blowups. Their resolution procedure (called the CJS algorithm) is ``canonical'' in a precise sense; in particular it commutes with localization and completion. The proof of termination is by contradiction, which is unusual in this business. The present paper introduces a new local invariant \(\iota(X, x)\) for schemes satisfying the above assumptions. In Theorem A, it is shown that this invariant decreases at each step of the CJS algorithm. This provides a new proof of termination of the CJS algorithm which is closer in spirit to the classical approach used in resolution theory. surface singularities; resolution of singularities; invariants for singularities; Hironaka's characteristic polyhedra Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A strictly decreasing invariant for resolution of singularities in dimension two

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author gives a unified, abstract treatment of the process of setting up a category \({\mathcal D}\) of geometric objects which -- like \(C^ \infty\)-manifolds, schemes or algebraic spaces -- are constructed by gluing together spaces from a (more) basic category \({\mathcal C}\) (the category of open subspaces of Euclidean space and \(C^ \infty\)-maps in the case of \(C^ \infty\)-manifolds, and the dual of the category of rings for the other two examples). The key ingredients qualifying the category \({\mathcal C}\) for such a local structure are: (i) A family Sub of distinguished maps called formal subsets, closed under composition and stable (the pullback of a formal subset exists and is a formal subset); the collection of formal subsets of each \(C \in {\mathcal C}\) is essentially small. Thus, schemes and algebraic spaces differ due to different choices of formal subsets in the category of affine schemes. (ii) For each \(C \in {\mathcal C}\) a collection \(\text{Cov} (C)\) of stable effective covers of \(C\) by formal subsets (epimorphic families of formal subsets which become pullback-stable colimiting cones for their canopies, the latter term referring to the diagram obtained from the domain-objects of a cover of \(C\) by filling in the projections of the obvious pairwise fibered products over \(C)\). The system Cov is required to satisfy certain axioms, amongst which are (essentially) those for a Grothendieck topology. A morphism of local structures or continuous functor preserves formal subsets, their pullbacks and the specified covers. A local structure \({\mathcal C}\) (equipped with Sub and Cov) becomes a global structure if each abstract canopy or ``cut-and-paste specification'' definable in it can be realized as that of a stable effective cover of some \(C \in {\mathcal C}\). The main result states that any local structure \({\mathcal C}\) can be universally completed to a global structure \({\mathcal D}\) via a fully faithful continuous functor \({\mathcal C} \to {\mathcal D}\) by iterating (twice) a certain ``plus-construction''. The classical examples (including rigid analytic spaces and Douady's espaces analytique banachique) fit in this mould, and can thus be usefully recognized as universal constructions. \(C^ \infty\)-manifolds; cut-and-paste specification; plus-construction; Banach analytic spaces; schemes; algebraic spaces; local structure; formal subsets; pullback-stable; canopies; Grothendieck topology; continuous functor; global structure; rigid analytic spaces; espaces analytique banachique; universal constructions Feit P., Axiomization of Passage from 'Local' Structure to 'Global' Object 91 pp 485-- (1993) Grothendieck topologies and Grothendieck topoi, Abstract manifolds and fiber bundles (category-theoretic aspects), Generalizations (algebraic spaces, stacks), Research exposition (monographs, survey articles) pertaining to category theory, Topoi, Banach analytic manifolds and spaces Axiomization of passage from ``local'' structure to ``global'' object

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system We give a new algorithm computing local system cohomology groups for complexified real line arrangements. Using it, we obtain several conditions for the first local system cohomology to vanish and to be at most one-dimensional, which generalize a result by Cohen-Dimca-Orlik. The conditions are described in terms of discrete geometric structures of real figures. The proof is based on a recent study on minimal cell structures. We also compute the characteristic variety of the deleted \(B_3\)-arrangement. complexified real line arrangements; minimal cell structures Yoshinaga, M, Resonant bands and local system cohomology groups for real line arrangements, Vietnam J. Math., 42, 377-392, (2014) Relations with arrangements of hyperplanes, Configurations and arrangements of linear subspaces, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Resonant bands and local system cohomology groups for real line arrangements

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The goal of the authors is to describe a detail procedure for the reduction of the multiplicity of a hypersurface singularity over an algebraically closed ground field of positive characteristic along a valuation under additional assumptions. One of them is a requirement that there exists a finite linear projection of the hypersurface singularity which is defectless. The authors underline that their main result (Theorem 7.1) follows also from the paper [\textit{J.-C. San Saturnino}, J. Algebra 481, 91--119 (2017; Zbl 1370.13005)]. It should be remarked that the paper under review contains an interesting brief historical background concerning the development of the classical theories of local uniformization and resolution of singularities originated by O. Zariski, a series of fruitful ideas, comments and useful references related to these topics. local uniformization; hypersurface singularities; resolution of singularities; defect; multiplicity; valuations; Perron transforms; Zariski's reduction of singularities; defectless projection Global theory and resolution of singularities (algebro-geometric aspects), General valuation theory for fields, Valuations, completions, formal power series and related constructions (associative rings and algebras) Defect and local uniformization

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\Lambda \) be a finitely generated associative algebra with unit over an algebraically closed field \(k,\) and let \(d\) be a fixed natural number. Each \( d\)-dimensional \(\Lambda \)-module corresponds to a unique \(r\)-tuple \(m=(m_{1},\dots ,m_{r}) \) of \(d\times d\) matrices over \(k\)., where \( \Lambda \) is generated over \(k\) by \(r\) elements. Through this identification the group \(\text{GL}_{d}(k) \) acts on the variety of \(d\)-dimensional module structures on \(k^{d}\) by conjugation. The orbits of this action are the isomorphism classes of \(d\)-dimensional \(\Lambda \)-modules. The \(\Lambda \)-module \(M\) is said to degenerate to the \(\Lambda \)-module \(N\) if \(n\) is in the Zariski closure of \(\mathcal{O}(m) ,\) the orbit of \(m\), where \(m\) and \(n\) correspond to \(M\) and \(N\) respectively. In this case the stabilizer in \(\text{GL}_{d}(k) \) of the point \(n\) acts on \(\mathcal{O}(m) \) by conjugation-- the orbit of any \( x\in \mathcal{O}(m) \) under this action is denoted \(\mathcal{O} _{n}(x).\) For each such \(r\)-tuple we define \(\text{End} (m) \) to be the set of all \(d\times d\) matrices (not necessarily invertible)\ which commute with \(m\). Clearly we can view \(\text{End} (m) \subset \text{End}_{k}(k^{d}).\) Given \(M\) and \(N\) such that \(M\) degenerates to \(N\) we say that \(N\) is a filtered degeneration of \(M\) if there exists an \(x\in \mathcal{O}(m) \) and \(y\in \mathcal{O}(n) \) such that \(\text{End}(x) \subset \text{End}(y) \subset \text{End} _{k}(k^{d}).\) This paper is a characterization of filtered degenerations in the case where \(\Lambda \) is the truncated polynomial ring \(k[ t] /(t^{r}).\) In this case the \(m\) corresponding to \(M\) is a \(d\times d\) matrix with \(m^{r}=0,\) furthermore the orbit of \(m\) is the set of matrices similar to \(m\). Thus \(m\) can be viewed as a direct sum of Jordan blocks \( J_{i}(0) \) with eigenvalue \(0\). Thus each orbit corresponds to a partition of \(d\) into positive integers. The result of the paper is as follows. Let \(M\) and \(N\) be \(d\)-dimensional \(k [ t] /(t^{r}) \)-modules such that \(N\) is a filtered degeneration of \(M\). Let \(m\) and \(n\) be the \(r\)-tuples of \(d\times d\) matrices corresponding to \(M\) and \(N\) as above. Then for each \(y\in \mathcal{O}(n) \) there exists some \(x_{y}\in \mathcal{O}(m) \) such that \(\dim \mathcal{O}_{y}(x_{y}) =\dim \mathcal{O}(m)-\dim \mathcal{O}(n) \) and for \(y^{\prime }\neq y\) \( \mathcal{O}_{y}(x_{y}) \) and \(\mathcal{O}_{y^{\prime }}(x_{y^{\prime }}) \) are disjoint.\ Also, \(\mathcal{O}(m) \) is the union of all the \(\mathcal{O}_{y}(x_{y}) \)'s. The authors conclude with the example when \(d=11\) and \(m\) corresponds to the matrix \(M=J_{7}(0) \oplus J_{4}(0).\) The partitions for seven different choices of \(N\) such that \(N\) is a filtered degeneration for \(M\) are given explicitly. module varieties; orbit closures; truncated polynomial rings Group actions on varieties or schemes (quotients), Representations of associative Artinian rings Degenerations of \(k[t]/(t^r)\)-modules giving stratification of the orbits

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The article under review is an introduction to the theory of algorithmic resolution of singularities of algebraic varieties and local principalization of sheaves of ideals. The authors explain the basic notions and discuss some natural properties of these algorithms, such as compatibility with pull-backs via smooth morphisms, equivariance, changes of base field, etc. They emphasize the importance of some fundamental ideas of Hironaka on the subject, such as his ``fundamental invariant'' (a certain fraction, involving the order of an ideal) and a notion of equivalence (requiring equalities of certain closed sets, which are singular loci). The authors present a fairly complete proof a an algorithmic resolution theorem (in characteristic zero), which is a streamlined version (and simplification) of one developed by Villamayor several years ago explained, for instance, in Chapter 6 of [\textit{S. D. Cutkosky}, Resolution of singularities. Graduate Studies in Mathematics 63. Providence, RI: American Mathematical Society (AMS). (2004; Zbl 1076.14005)]. Singularities; resolution of singularities; equivalence; basic object; log-principalization; equivariance Benito, A., Encinas, S., Villamayor, O.: Some natural properties of constructive resolution of singularities. Asian J. Math. 15, 141-192 (2011) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Some natural properties of constructive resolution of singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system First we describe the setting for this paper. Let \(I\) be an ideal of the polynomial ring \(R=k[X_1,\dots,X_n]\) over a field \(k\). Assume: \(\bullet\) \(I\) is generated by \(n+1\) \(k\)-linearly independent forms \( \mathbf f=(f_0,\dots,f_n)\) of the same degree \(>0\); \(\bullet\) \(\dim (R/I)\leq 1\) (that is, the base locus of the rational map defined by \(\mathbf f\) consists of finitely many points); \(\bullet\) \(\dim k[\mathbf f]=n\) (that is, the image of the rational map defined by \(\mathbf f\) is a hypersurface). Recall that the kernel of the canonical epimorphism \(\text{Sym}_R(I))\to \text{Rees}_R(I)\) is equal to the \(R\)-torsion of the symmetric algebra \(\text{Sym}_R(I)\); hence it contains \(H_{\mathfrak m}(\text{Sym}_R(I))\) (the \(\mathfrak m\)-torsion of the algebra \(\text{Sym}_R(I))\). Thus we have an induced epimorphism \[ S_I^*\twoheadrightarrow \text{Rees}_R(I), \] where \(S_I^*=\text{Sym}_R(I)/H_{\mathfrak m}(\text{Sym}_R(I))\). In some particular cases, for example if \(I\) is \(\mathfrak m\)-primary, this epimorphism is an isomorphism. On the other hand, if \(S\) is the polynomial ring \(R[T_0,\dots, T_n]\), then there is an \(R\)-algebra graded homomorphism \(S\to S_I^*\) sending each \(T_i\) to \(f_i\). Let \(\mathcal J\) be the kernel of this homomorphism. For an integer \(\ell\geq0\), let \(\mathcal J(\ell)\) be the ideal generated by the elements in \(\mathcal J\) of total degree in the \(T_i\)'s at most \(\ell\). The basic numerical invariant of the paper is \textit{the treshold degree} of the ideal \(I\) denoted by \(\mu_0(I)\), which is defined using, among other things, the first homology module of the Koszul complex associated to \(\mathbf f\). A \textit{treshold integer} is an integer \(\mu\) such that \(\mu\geq\mu_0(I)\), where \(\mu_0(I)\). The main theorem of this paper (using the above setting) is the following: If \(\nu(I_{\mathfrak p})\leq \dim R_{\mathfrak p}+1 \) for every prime ideal containing \(I\), then for every treshold integer \(\mu\), we have \(H_{\mathfrak m}^i(\text{Sym}_r(I))_\mu=0\) for \(i>0\); and for every \(\ell\geq2\) the \(k[T_0,\dots,T_n]\)-module \((\mathcal J(\ell)/\mathcal J(\ell -1))_\mu \) if free. Moreover, the authors determine the rank of this free module. In particular, the authors deal with the case when the ideal \(I\) is \(\mathfrak m\)-primary. They use \textit{downgrading maps}; as mentioned in the paper, by this method, the equations of the Rees algebra of \(I\) used in the matrix-based representation of the corresponding hypersurface can be recovered from the syzygies of \(I\). The authors present some applications to the hypersurface implicitization problem, and include a detailed explicit example. Rees algebras; symmetric algebras; rational maps; elimination; Koszul complex; downgrading map; implicitization Busé, L.; Chardin, M.; Simis, A., Elimination and nonlinear equations of Rees algebras, J. algebra, 324, 6, 1314-1333, (2010) Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Syzygies, resolutions, complexes and commutative rings, Hypersurfaces and algebraic geometry Elimination and nonlinear equations of Rees algebras

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This monograph presents a local theory of planar and space curve singularities both from an algebraic and geometric point of view, being motivated by possible extensions of the Zariski equisingularity theory to the case of space curves and by making links to the theory of arc spaces, resolution of singularities, valuations. The book covers the following topics. The first chapter is devoted to branches at points, defined in several different ways and characterized by parameterizations, notably the Hamburger-Noether and Puiseux expansions. The second chapter treats the geometric features and numerical invariants extracted from local rings, and among them semigroup of values, Arf closures (multiplicity sequences by successive blow-ups), saturation (invariants of plane projection), including their computation through parameterizations. The third chapter introduces sequences of infinitely near points and their representation via Hamburger-Noether matrices. In chapters four and five, infinitely near points are studied geometrically by means of the divisorial theory of singularities. Many results hold over arbitrary perfect fields, which makes sense from the arithmetic and computational point of view. In general, the presentation is clear and self-contained, and satisfies most of readers' requirements. planar and space curve singularities; branches and parameterizations; Hamburger-Noether matrices; resolution of singularities; infinitely near points Campillo, A., Castellanos, J.: Curve Singularities. An Algebraic and Geometric Approach. Actualités Mathématiques, 126pp. Hermann, Paris (2005) Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of algebraic curves, Modifications; resolution of singularities (complex-analytic aspects) Curve singularities. An algebraic and geometric approach

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(X\) be a smooth curve over a finite field of characteristic \(p\), let \(E\) be a number field, and let \(\mathbf{L} = \{\mathcal{L}_\lambda\}\) be an \(E\)-compatible system of lisse sheaves on the curve \(X\). For each place \(\lambda\) of \(E\) not lying over \(p\), the \(\lambda\)-component of the system \(\mathbf{L}\) is a lisse \(E_\lambda\)-sheaf \(\mathcal{L}_\lambda\) on \(X\), whose associated arithmetic monodromy group is an algebraic group over the local field \(E_\lambda\). We use Serre's theory of Frobenius tori and \textit{L. Lafforgue}'s proof [Invent. Math. 147, 1--241 (2002; Zbl 1038.11075)] of Deligne's conjecture to show that when the \(E\)-compatible system \(\mathbf{L}\) is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of~\(\lambda\)''. More precisely, after replacing \(E\) by a finite extension, there exists a connected split reductive algebraic group \(G_0\) over the number field \(E\) such that for every place \(\lambda\) of \(E\) not lying over~\(p\), the identity component of the arithmetic monodromy group of \(\mathcal{L}_\lambda\) is isomorphic to the group \(G_0\) with coefficients extended to the local field \(E_\lambda\). C. Chin, Independence of \(\ell\) of monodromy groups, J. Amer. Math. Soc. 17 (2004), 723--747. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Étale and other Grothendieck topologies and (co)homologies Independence of \(\ell\) of monodromy groups

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In this paper, there is a new proof of the existence and a construction of a resolution of excellent schemes of finite type over a ground field of characteristic 0 and of principalization of an ideal of a smooth scheme. This proof is based on the earlier works of Villamayor, Villamayor-Encinas and Bierstone-Milman, all working after Hironaka. The main point is as usual the ``maximal contact''. What is new~? \noindent 1. This proof is concise (14 pages). 2. The Hilbert-Samuel function is avoided. The proof starts as usual by the construction of a ``basic object'' \((W_0, (J_0,b),E)\) where \(W_0\) is a smooth variety, \(J_0\subset{\mathcal O}_{W_0}\) an ideal, \(b\in \mathbb N\), \(E_0\) is a normal crossing divisor of \(W_0\). The proof is in two parts: \noindent 1- Resolve any basic object, \noindent 2- Prove that the resolution of basic objects leads to resolution of embedded schemes and to principalization of ideals. In the case of desingularization, the main invariant used is the order of \(J_0\) restricted to \(W_0\) a smooth variety of maximal contact. The strata given by this order are not the Hilbert-Samuel strata. This is an improvement: these new strata can be computed easily, which is not the case for the H-S strata. Hence, this new algorithm is easier to implement. A problem arises: the authors do not look at the strict transform of \({J_0}\), but at its weak transform \(t^{-b}{J_0}\) where div\((t)\) is the exceptional divisor of the blowing-up. This small modification with the usual proofs simplifies the redaction but, then, it is not clear at all that the algorithm is independent of the embedding: the weak transform \(t^{-b}{J_0}\) depends obviously on the embedding and on the choice of \(W_0\). This difficulty is easily overcome: the authors show that two different \(W_0\) and \(W'_0\) have same dimension and that you can find an étale covering \(\tilde W_0\) where the pull back of the basic objects on \(W_0\) and \(W'_0\) are the same, so the algorithms coincide on each pair \((W_0, \tilde W_0)\) and \((W'_0, \tilde W_0)\). This means that the authors' construction has many properties of invariance that should be interpreted geometrically. Encinas S., Villamayor O.: A proof of desingularization over fields of characteristic zero. Rev. Mat. Iberoamericana 19(2), 339--353 (2003) Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) A new proof of desingularization over fields of characteristic zero

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R,\mathfrak{m})\) be a regular local ring, \(J\subset R\) a non-zero ideal and \(u=(u_1,\ldots,u_d)\) a system of regular elements that can be extended to a regular system of parameters of \(R\). To this data Hironaka associates the characterisctic polyhedron \(\Delta(J;u)\) (see [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)]). It is proved that the polyhedron depends only on \(R/J\) and \(u\) mod \(J\). This implies that numerical data obtained from the polyhedron are invariants of the singularity \(R/J\). invariants for singularities; Hironaka's characteristic polyhedra; resolution of singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polytopes and polyhedra, Modifications; resolution of singularities (complex-analytic aspects) Invariance of Hironaka's characteristic polyhedron

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R=F_q [\mathbf{y}]=F_q [y_1,\dots,y_n]\) be a polynomial ring over a finite field \(F_q\) and let \(y^{v_1},\dots,y^{v_s} \) be a finite set of monomials in \(F_q [\mathbf{y}]\). The affine and projective spaces over the field \(F_q\) of dimensions \(s\) and \(s - 1\) by \(A^s\) and \(P^{s-1}\), respectively. Let \(X\) be the set consisting of all points \([(y^{v_1},\dots,y^{v_s})] \) in \(P^{s-1}\) that are well defined, i.e., \(x\in F_q^n\) and \(x^{v_i} \neq 0 \) for some \(i\). \(X\) is called of \textit{clutter type} if \(\mathrm{supp}(y^{v_i})\not\subseteq\mathrm{supp}(y^{v_j})\) for \(i\neq j\), where \(\mathrm{supp}(y^{v_i})\) is the support of the monomial \(y^{v_i}\) consisting of the variables that occur in \(y^{v_i}\). Let \(S = F_q[t_1,\dots,t_s] = \oplus_{d=0}^{\infty} S_d\) be a polynomial ring over the field \(F_q\) with the standard grading. The graded ideal \(I(X)\) generated by the homogeneous polynomials of \(S\) that vanish at all points of \(X\) is called the vanishing ideal of \(X\). The vanishing ideal \(I(X)\) is a complete intersection if \(I(X)\) is generated by \(s-1\) homogeneous polynomials. In this paper, the authors give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields. binomial ideal; complete intersection; finite field; monomial parameterization; vanishing ideal Tochimani, A.; Villarreal, R. H.;, Vanishing ideals over rational parameterizations, arXiv:1502.05451v1, (2015) Complete intersections, Finite ground fields in algebraic geometry, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Algebraic coding theory; cryptography (number-theoretic aspects) Complete intersection vanishing ideals on sets of clutter type over finite fields

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This second edition (see [Zbl 1308.14001] for a review of the first edition) is extended by a chapter on recent developments. After a short motivating first chapter that introduces the questions addressed in this book, follow two chapters dealing with background, one on sheaves, algebraic varieties and analytic spaces, and one on homological algebra and duality. The treatment includes spectral sequences. In these chapters the theorems are formulated in their natural generality. The definition of a singularity is initially given both for analytic spaces and for algebraic varieties over an arbitrary algebraically closed field. After stating Artin's Algebraization Theorem, that an isolated singularity of an analytic space is isomorphic to the germ of an algebraic variety over \(\mathbb{C}\), only the algebraic case is considered. After a chapter defining the canonical divisor for varieties over an arbitrary algebraically closed field the further discussion is restricted to the field of complex numbers. The book defines log canonical, canonical, log terminal, terminal and rational singularities and provides a characterization of isolated such ones in terms of plurigenera. The classification is refined in the two-dimensional case, and rational surface singularities are described in some detail. Also the results of the Author on two-dimensional Du Bois singularities are introduced. The next chapter considers the analogous questions for higher dimensional singularities, and in particular for the case of dimension three. It concludes with the list of the famous 95 families of simple \(K3\)-singularities. The final chapter presents some developments after the publication of the first Japanese version [Zbl 1308.14002] of this book. These concern log discrepancies for pairs and the use of the space of arcs in their description. This opens up the possibility of proving results in positive characteristic. The book closes with a brief introduction to \(F\)-singularities, in positive characteristic. rational singularities; minimal model program; Du Bois singularities; arc spaces; F-singularities Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Introduction to singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The problem of resolving the singularities of an algebraic variety by a sequence of birational transformations has a long history. A ground- breaking step forward was made in 1940, when O. Zariski developed an ingenious method for uniformizing hypersurface singularities (in characteristic zero) by a process of successive substitutions of variables in polynomial or power series [cf. \textit{O. Zariski}, Ann. Math., II. Ser. 41, 852-896 (1940; Zbl 0025.21601)]. Then, in 1964, H. Hironaka gave an affirmative answer to the whole problem of resolving singularities in characteristic zero, essentially by generalizing Zariski's approach to a general process of successive ``permissible'' blow-up transformations, expressible in the full scheme-theoretic framework [cf. \textit{H. Hironaka}, Ann. Math., II. Ser. 79, 109-326 (1964; Zbl 0122.386)]. After that celebrated paper of Hironaka's, many attempts have been undertaken to analyse the constructiveness of his process of desingularization in various concrete situations. Among them are the papers of \textit{S. S. Abhyankar} [cf. ``Weighted expansions for canonical desingularization'', Lect. Notes Math. 910 (1982; Zbl 0479.14009)], \textit{O. E. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], \textit{E. Bierstone} and \textit{P. Milman} [J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007)] and others. In the present paper, the author provides another approach to the presentation and uniformization of hypersurface singularities. Generalizing Zariski's method and systematizing Hironaka's ``quasi- canonical resolution procedure'' for hypersurface singularities with a normal crossing factor, he constructs a numerical sequence for any hypersurface singularity, which classifies the singularity completely and, moreover, describes a permissible resolution procedure in a very concrete and effective way. As the author points out, his systematized approach has the advantage of being applicable to the study of hypersurface singularities in positive characteristic, too [cf. the author, Publ. Res. Inst. Math. Sci. 23, No. 6, 965-973 (1987; Zbl 0657.14002)]. resolving the singularities; uniformization of hypersurface singularities [M]Moh, T. T., Canonical uniformization of hypersurface singularities of characteristic zero.Camm. Algebra 20 (1992), 3207--3251. Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Quasi-canonical uniformization of hypersurface singularities of characteristic zero

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors describe an algorithm to determine a minimal presentation of the canonical module \(\omega\) of the coordinate ring \(R\) of a set of points \(X\) of the projective space \(\mathbb{P}^n\). The approach is based on the computation of a system of generators of \(\omega\) which is also a Gröbner basis with respect to a suitable filtration. The authors also explain how the knowledge of a minimal presentation of \(\omega\) can be used to speed up the computation of the minimal free resolution of \(R\). They use this construction to prove or to give counterexamples to the minimal resolution conjecture of \textit{A. Lorenzini} [J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)] for small \(n\) and \(| X|\). minimal presentation of the canonical module of the coordinate ring; set of points of projective space; Gröbner basis S. Beck and M. Kreuzer, ``How to compute the canonical module of a set of points'' in Algorithms in Algebraic Geometry and Applications (Santander, Spain, 1994) , Progr. Math. 143 , Birkhäuser, Basel, 1996, 51--78. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Relevant commutative algebra, Symbolic computation and algebraic computation How to compute the canonical module of a set of points

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Given a very ample divisor \(H\) on a smooth projective algebraic surface \(S\) over a field of characteristic \(0\), the authors give an explicit lower bound to \(d\) such that a collection of isolated singularities with prescribed topological types, located in \(S\) in general position, imposes independent conditions to the linear system \(|dH|\). More precisely, with an isolated singular curve point one can associate a zero-dimensional scheme (for instance, the scheme associated with an ordinary \(m\)-fold point is defined by the \(m\)-th power of the maximal ideal in the local ring of the surface), and the main theorem claims that \(H^1(S,{\mathcal J}(d))=0\) for the ideal sheaf \({\mathcal J}\) of a ``generic'' zero-dimensional scheme associated with the given multi-singularity. The bound to \(d\) becomes sharper when most of singularities are ordinary nodes. The proof is based on the so-called ``Horace method'', invented by \textit{A. Hirschowitz} [Manuscripta Math. 50, 337-388 (1985; Zbl 0571.14002)]. Under some extra conditions, there exists an irreducible curve in \(|dH|\) having exactly the prescribed collection of singularities. Reviewer's remark. Another approach to the problem of existence of curves with prescribed singularities in given linear systems on smooth algebraic surfaces, based on the Kodaira vanishing theorem, was suggested by \textit{T. Keilen} and \textit{I. Tyomkin} [Trans. Am. Math. Soc. 354, No. 5, 1837-1860 (2002; Zbl 0996.14013)]. singular curves; zero-dimensional schemes; vanishing of cohomology; Horace method Singularities of curves, local rings, Vanishing theorems in algebraic geometry, Singularities in algebraic geometry Imposing singular points and many nodes to curves on surfaces

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(G\) be a split reductive group over a field of characteristic \(p\) with a chosen maximal torus \(T\) and Levi subgroup \(L\). Let \(\Lambda^+\) be a set of dominant weights, where dominance is defined using a fixed Borel subgroup of \(G\) containing \(T\). Additionally, let \(\check{G}\) be the connected complex algebraic group with maximal torus \(\check{T}\) with root datum dual to \(G\). The geometric Satake theorem establishes a equivalence between the category of rational representations of \(G\) and the category \(P(\mathcal{G}r)\) of \(\check{G}(\mathcal{C}[[t]])\)-equivariant perverse sheaves on \(\mathcal{G}r\), the affine Grassmannian for \(\check{G}\) with coefficients on \(k\). If \(p\) is not a torsion prime for \(\check{G}\), then for each \(\lambda\in \Lambda^+\) one can assign a certain indecomposable parity complex \(\mathcal{E}(\lambda)\): these particular parity complexes are called parity sheaves. Call a \(\mathcal{T}\in P(\mathcal{G}r)\) a titling sheaf if it corresponds to a tilting module for \(T\). The aim of the paper being reviewed is to describe a connection between parity sheaves and tilting modules, one which explains the similarities between their properties. The main result is that, provided \(p\) is large enough, \(\mathcal{E}(\lambda) = \mathcal{T}(\lambda)\) for all \(\lambda\in \Lambda^+\). The precise definition of ``large enough'' depends on the type of the root system \(\Phi\) for \(G\), however one can always take \(p>n\) is \(\Phi\) is of type \(C_n\); otherwise one can apply a lower bound \(p> 31\) (a bound which can be further refined). perverse sheaves; tilting modules Juteau, D.; Mautner, C.; Williamson, G., Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér. (4), 49, 2, 257-275, (2016) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Classical groups (algebro-geometric aspects), Geometric Langlands program: representation-theoretic aspects, Representations of Lie and linear algebraic groups over local fields Parity sheaves and tilting modules

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper initiates the author's Diophantine study of modulus spaces for local systems on surfaces and their mapping class group dynamics. The aim of the paper is to establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems. This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the author collects relevant background on surfaces and moduli of local systems. Section 3, deals with systoles of local systems. In this section, as a first step towards the main result, the author introduces the notion of systole for \(SL_2\)-local systems, and studies its boundedness properties. In Section 4, the author proves a compactness criterion for local systems, which extends Mumford's compactness criterion [\textit{D. Mumford}, Proc. Am. Math. Soc. 28, 289--294 (1971; Zbl 0215.23202)] on moduli of closed Riemann surfaces as well as a related work of [\textit{B. H. Bowditch} et al., Math. Ann. 302, No. 1, 31--60 (1995; Zbl 0830.57008)]. Section 5, deals with some remarks. Here the author collects further remarks on the main result, briefly visits the theory of arithmetic hyperbolic surfaces and derives elementary observations on the behavior of integral points on the moduli spaces \(X_k\). He gives an alternative proof of the finitude of mapping class group orbits for non degenerate faithful representations in \(X_k(\mathbb{Z})\) for surfaces with boundary. modulus spaces; local systems; systoles; Riemann surfaces Algebraic moduli problems, moduli of vector bundles, Group actions on varieties or schemes (quotients), 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.), Cubic and quartic Diophantine equations, Rational points Nonlinear descent on moduli of local systems

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper under review represents a continuation of the authors' work on the interpolation problem posed by a finite number of general points with weights in the projective plane, i.e., the problem of determining the dimension of the linear system \({\mathcal L}_d(p_1^{m_1},\dots,p_n^{m_n})\) of plane curves of degree \(d\) with prescribed multiplicity at prescribed general points in the projective plane. By the Riemann-Roch Theorem, such a linear system has the expected dimension if \(H^1(S,{\mathcal L})=0\), where \({\mathcal L}\) is the line bundle corresponding to \({\mathcal L}_d(p_1^{m_1},\dots,p_n^{m_n})\) on the blow-up \(S\) of the plane at the prescribed points. Segre's Conjecture states that if the linear system \(|{\mathcal L}|\) is reduced, meaning it is nonempty and a general curve in \(|{\mathcal L}|\) has at most isolated singularities, then \(H^1(S,{\mathcal L})=0\). The paper's main result is a proof of Segre's Conjecture for the case of all weights being at most three. The proof is essentially a deformation theory argument. \(H^1(S,{\mathcal L})\) is shown to be an obstruction space for deformations of a general element of \(|{\mathcal L}|\) as the points \(p_i\) vary. However, as the \(p_i\) are general, there are no obstructions, and one can conclude \(H^1(S,{\mathcal L})=0\). Divisors, linear systems, invertible sheaves, Plane and space curves, Rational and ruled surfaces A deformation theory approach to linear systems with general triple points

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(I\) be an ideal \(I\subset S = k[x_1,\dots,x_r]\) such that \(A= S/I\) is Artinian; then \(A\) is said to have the Weak Lefschetz Property (WLP) if \(\exists l\in S_1\) such that \(\forall m\) the multiplication map \(\mu_l:A_m \rightarrow A_{m+1}\) has full rank. Notice that if such an \(l\) exists, then for a generic form \(l\in S_1\), \(\mu_l\) has full rank. Previous results show that WLP always holds for \(r\leq 2\), while for \(r=3\) it holds in many cases, e.g. when \(I\) is generated by powers of linear forms. This last result fails for \(r\geq4\). In this paper WLP of \(A\) is studied in general when \(I\) is generated by powers of linear forms; by the well known correspondence (via inverse systems) between such ideals and ideals of fat points, the problem can be studied from a geometric point of view, and several results are proved, namely: - If \(I=(l_1^t,\dots,l_n^t)\), where the \(l_i\)'s are general in \(S_1\), the map \(\mu_l\) has full rank if and only if \((r,t,n) \notin \{(4,3,5),(5,3,9),(6,3,14),(6,2,7)\}\) (this depends on Alexander-Hirschowitz classification of irregular linear system defined by 2-fat points in \({\mathbb P}^n\)). - Let \(I \in k[x_1,\dots,x_4]\), be generated by general linear forms. For \(n\in \{5,6,7,8\}\) the WLP fails for, respectively, \(t\geq 3,27,140,704\). - For \(n=r+1\) and \(r=2k\geq 4\), WLP fails in degree \({r \over 2}(t-1)-1\), for \(t\gg 0\). By using Gelfand-Tsetlin patterns, partial results are obtained to extend the last result for \(r\) odd, thus suggesting the following conjecture: \[ \text{For \(n \geq r+1\geq 5\), WLP fails for \(t\gg 0\).} \] inverse systems; powers of linear forms; fat points; Lefschetz property R. M. Miró-Roig, Ordinary curves, webs and the ubiquity of the Weak Lefschetz Property, Algebras and Representation Theory (2014) to appear. 10.1007/s10468-013-9460-9. Syzygies, resolutions, complexes and commutative rings, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Other special types of modules and ideals in commutative rings, Linkage, complete intersections and determinantal ideals, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a commutative Noetherian ring with non-zero identity, \(\mathfrak{a}\) an ideal of \(R, M\) a finitely generated \(R\)-module, and \(a_1,\dots, a_n\) an \(\mathfrak{a}\)-filter regular \(M\)-sequence. The formula \[\text{H}^{i}_\mathfrak{a} (M)\cong \begin{cases} \text{H}^{i}_{(a_1,\dots,a_n)}(M) & \text{for all } i<n, \\ \text{H}^{i-n}_{\mathfrak{a}}(H^{n}_{(a_1,\dots,a_n)}(M)) & \text{for all } i\geq n, \end{cases}\] is known as Nagel-Schenzel formula and is a useful result to express the local cohomology modules in terms of filter regular sequences. In this paper, we provide an elementary proof to this formula. filter regular sequences; local cohomology modules; Nagel-Schenzel formula Local cohomology and commutative rings, Local cohomology and algebraic geometry An elementary proof of Nagel-Schenzel formula

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be an algebraic group defined over a number field \(K\), let \(P\) be a point in \(A(K)\), and let \(G\) be a finitely generated subgroup of \(A(K)\). If \(P\) belongs to \(G\), then clearly its reduction \((P\bmod\mathfrak{p})\) belongs to \((G\bmod\mathfrak{p})\) for all but finitely many primes \(\mathfrak{p}\) of \(K\) (notice that we only consider those primes \(\mathfrak{p}\) such that the reductions are well-defined, and are ``good'' reductions). The problem of detecting linear dependence asks whether the converse holds, so whether we have a local-global principle. In this survey article we also investigate the problem of detecting linear dependence for torsion points. number fields; local-global principle; detecting linear dependence Abelian varieties of dimension \(> 1\), Group varieties, Arithmetic ground fields for abelian varieties The problem of detecting linear dependence

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(\mathbb{A}^ N\) be affine space with coordinates \((t_ 1,\dots,t_ N)\), and \(z_ 1,\dots,z_ n\) fixed real numbers. Let \(U\) be the complement of all hyperplanes \(t_ p = z_ q\), \(t_ p = t_ q\) in \(\mathbb{A}^ N\) and \(j: U \to \mathbb{A}^ N\) the inclusion. Let \(\mathcal L\) be a local system over \(U\) whose monodromy around hyperplanes \(t_ p = z_ q\) (resp. \(t_ p = t_ q\)) is \(\exp(-2\pi i(\alpha_ p,\Lambda_ q)/k)\) (resp. \(\exp(2\pi i(\alpha_ p,\alpha_ q)/k))\), where \(\Lambda_ 1,\dots,\Lambda_ n\), \(\alpha_ 1,\dots,\alpha_ r\) are vectors in the dual space of a finite-dimensional vector space \(\mathfrak h\) with nondegenerate symmetric bilinear form ( , ), \(k\) is a nonzero complex number. Complexes \(j_ !{\mathcal L}[N]\), \(Rj_ *[N]\) are perverse sheaves over \(\mathbb{A}^ N\) smooth along the stratification defined by the above hyperplanes. A construction which associates to any such sheaf a quiver is considered. The method of calculating the cohomology of a perverse sheaf in terms of its quiver is presented. Standard quivers based on a complex of a free Lie algebra with coefficients in tensor powers of its universal enveloping algebras are constructed. The main result of the paper says that these quivers are ``quasi-classical'' limits of quivers associated to \(j_ !{\mathcal L}[N]\), \(Rj_ *{\mathcal L}[N]\). Most of the results are only formulated without proofs. Hochschild homology; quasi-classical quiver; sheaves; cohomology Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Representation theory of associative rings and algebras Vanishing cycles and quantum groups. I

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(S\) be a smooth surface and let \(L\) be a linear system on \(S\), whose general element is smooth and irreducible. The \(\delta\)-th Severi variety \(V\) of \(L\) parameterizes curves in \(|L|\) which are nodal, irreducible, with \(\delta\) nodes for singularities. The authors study the case of Enriques surfaces \(S\), with the involution \(\pi:X\to S\) from the universal covering. The target is the determination of the codimension of components of the \(\delta\)-th Severi variety \(V\) in \(|L|\) at a fixed nodal curve \(C\). It turns out that the codimension depends on the lifting \(\pi^{-1}(C)\) of \(C\) in the involution. If the lifting is irreducible, then the codimension has the naïvely expected value \(\delta\). On the other hand, if the lifting splits in two irreducible components, then the authors prove that the codimension of \(V\) at \(C\) is \(\delta-1\). Enriques surfaces; nodal curves \(K3\) surfaces and Enriques surfaces A note on Severi varieties of nodal curves on Enriques surfaces

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This is a survey of the main results from the authors' five papers on the cohomology of local systems on complements of complex hyperplane arrangements, culminating in explicit description and analysis of the Gauss-Manin connections associated with cohomology of rank-one local systems. Let \(\mathcal A\) be such an arrangement, with complement \(M.\) The starting point is a construction by the first author of a weakly self-indexing, perfect Morse function relative to the Whitney stratification of \({\mathbb C}^\ell\) defined by \(\mathcal A\). This is first used to give a short proof (found independently by \textit{M.~Yoshinaga} [Kodai Math. J. 30, No. 2, 157--194 (2007)]) of the result of Randell and Dimca-Papadima that \(M\) has a minimal cell decomposition. The authors proceed to construct a universal finite-dimensional complex which computes the cohomology, via specialization, of an arbitrary local system on \(M.\) There is a natural linearization of this complex at the trivial local system which for rank-one local systems is chain equivalent to the Aomoto complex of \({\mathcal A}.\) As a consequence the tangent cones at \(1\) to the characteristic varieties of \(M\) in \(({\mathbb C}^*)^n\), the jumping loci for cohomology of rank-one local systems, coincide with the corresponding resonance varieties of \(\mathcal A\) in \({\mathbb C}^n\), the jumping loci for the cohomology of the Orlik-Solomon algebra of \(\mathcal A.\) The authors then consider the space \(B({\mathcal T})\) of all arrangements of fixed combinatorial type \({\mathcal T}\), and the bundle over \(B({\mathcal T})\) of cohomology groups of local systems determined by the parameter \(t=\exp(-2\pi i\lambda)\in ({\mathbb C}^*)^n.\) The monodromy of this bundle is the monodromy of a flat connection, the Gauss-Manin connection, described by a collection of linear maps \(\Omega({\mathcal T},{\mathcal T}')\) where \({\mathcal T}'\) is a codimension-one degeneration of \({\mathcal T}.\) Following \textit{H.~Terao} [Topology Appl. 118, No. 1--2, 255--274 (2002; Zbl 1020.52020)], the authors give a formula for \(\Omega({\mathcal T},{\mathcal T}')\) in terms of the Gauss-Manin connection for general position arrangements, which were computed by Aomoto and Kita. The matrices \(\Omega({\mathcal T},{\mathcal T}')\) are written explicitly using nbc and \(\beta\)nbc bases. It is shown that the entries of \(\Omega({\mathcal T},{\mathcal T}')\) are linear in \(\lambda,\) and that \(\Omega({\mathcal T},{\mathcal T}')\) is diagonalizable with at most one nonzero eigenvalue (under a weak non-resonance assumption). All of this is illustrated in the last section with detailed calculations for the rank-three Selberg arrangement. arrangement of hyperplanes; local system cohomology, Gauss-Manin connection, stratified Morse theory Cohen, D; Orlik, P, Stratified Morse theory in arrangements, Pure Appl. Math. Q. 2, 3, 673-697, (2006) Relations with arrangements of hyperplanes, Structure of families (Picard-Lefschetz, monodromy, etc.), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology, Discriminantal varieties and configuration spaces in algebraic topology Stratified Morse theory in arrangements

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A root system \(R\) of rank \(n\) defines an \(n\)-dimensional smooth projective toric variety \(X(R)\) associated with the fan of Weyl chambers. Various properties of varieties \(X(R)\) have been studied in work of \textit{I. Dolgachev} and \textit{V. Lunts} [J. Algebra 168, No. 3, 741--772 (1994; Zbl 0813.14040)], \textit{A. A. Klyachko} [Funct. Anal. Appl. 19, 65--66 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 77--78 (1985; Zbl 0581.14038)], \textit{C. Procesi} [Langue, Raisonnement, Calcul, 153--161 (1990; Zbl 1177.14090)], \textit{J. R. Stembridge} [Adv. Math. 106, No. 2, 244--301 (1994; Zbl 0838.20050)]. This article is inspired by the article of \textit{A. Losev} and \textit{Yu. Manin} [Aspects of Mathematics E 36, 181--211 (2004; Zbl 1080.14066)], where the fine moduli spaces \(\overline{L}_n\) were constructed and it was shown that \(\overline{L}_{n+1}\) coincides with \(X(A_n)\). The Losev-Manin moduli space \(\overline{L}_n\) parametrises the isomorphism classes of chains of projective lines with two poles and \(n\) marked points. In this article, the authors consider functorial properties of \(X(R)\) with respect to maps of root systems and propose a description of the functor of toric varieties \(X(R)\) that is based on projection maps \(X(R) \to X(A_1) \simeq \mathbb P^1\). In the special case of \(R=A_n\), the author prove that \(X(A_n)\) is an almost Fano variety and the anticanonical divisor defines a birational toric morphism to the Gorenstein toric Fano variety corresponding to the convex hull of all roots of \(A_n\). A new proof is given for the fact that the toric varieties \(X(A_n)\) are fine moduli spaces \(\overline{L}_{n+1}\). toric varieties; root systems; Losev-Manin moduli spaces Batyrev, V.; Blume, M., The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces, Tohoku Math. J. (2), 63, 581-604, (2011) Toric varieties, Newton polyhedra, Okounkov bodies, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The authors study plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. Before we formulate some results for Cremona maps (including the main result for a plane Cremona map of degree \(d \geq 4\)) we need to recall some definitions. Assume that \(k\) is an infinite field. A linear system of plane curves of degree \(d\) is a \(k\)-vector subspace \(L_{d}\) of the vector space of forms of degree \(d\) in the standard graded polynomial ring \(R := k[x,y,z]\). A linear system \(L_{d}\) defines a rational map \(\mathcal{L}_{d} : \mathbb{P}^{2} \rightarrow \mathbb{P}^{r}\), where \(r+1\) is the vector space dimension of \(L_{d}\). If a map is birational, then it is called a Cremona map. Write \(I = (I : F)F\), where \(F \in R\) is, up to non-zero scalars, a uniquely defined form of degree \( \leq d\) such that \(I : F\) has codimension greater or equal to two. Then \(F\) is the fixed part of the system. We say that the linear system has no fixed part meaning that \(\text{deg} \,F = 0\). Here in the paper the authors assume that \(I : F\) has codimension equal to two. Given a variety \(X\), a smooth point \(P\) and a divisor \(D\) then the multiplicity \(e_{P}(D)\) is defined as \[ e_{P}(D) = \text{max} \{s \geq 0 : f \in \mathfrak{m}_{P}^{s}\}, \] where \(f\) is a local equation of \(D\) and \(\mathfrak{m}_{P}\) is the maximal ideal of the local ring of \(X\) at \(P\). The virtual multiplicity of \(L_{d}\) at one of its proper base points \(P\) is defined as \[ \mu_{P} = \mu_{P}(L_{d}) : = \text{min}\{e_{P}(f) : f \in L_{d}\}. \] An important property of a plane Cremona maps is that its virtual multiplicities satisfy the classical equations of condition \[ \sum_{P} \mu_{P} = 3d-3, \,\,\,\, \sum_{P} \mu_{P}^{2} = d^{2} - 1, \] where \(P\) runs over the set of base points of the corresponding \(L_{d}\) with respect to multiplicities \(\mu_{P}\). An abstract configuration \((d, \mu_{1},\dots, \mu_{r})\) satisfying the equations of condition is called a homaloidal type. A homaloidal type is called proper if there exists a plane Cremona map with this type. A de Jonquiéres map is a plane Cremona map \(\mathfrak{F}\) of degree \(d\geq 2\) having the homaloidal type \((d; d-1, 1^{2d-2})\). Now we are given a set \(\mathcal{P} = \{P_{1}, \dots, P_{n} \} \subset \mathbb{P}^{2}\) of points with multiplicities \((\mu_{1}, \dots, \mu_{n})\). The \(\mu\)-fat ideal of \(\mathcal{P}\) is given by \[ I(\mathcal{P}; \mu) = \bigcap_{i=1}^{n} I(P_{i})^{\mu_{i}}, \] where \(I(P_{i})\) is the homogeneous prime ideal of the point \(P_{i}\). Recall that a net of degree \(d\) is a linear system \(L_{d}\) spanned by three independent forms in \(R_{d}\) without a proper common factor. We say that the net defined by \(L_{d}\) is complete if \(L_{d} = J_{d}\), where \(J = I(\mathcal{P}; \mu)\). At last, the net \(L_{d}\) is homaloidal if it defines a Cremona map of \(\mathbb{P}^{2}\). Now we are ready to present the main result of the paper. Theorem. Let \(\mathfrak{F} : \mathbb{P}^{2} \dashrightarrow \mathbb{P}^{2}\) be a Cremona map with homaloidal type \((d; \mu)\), with \(d \geq 4\), whose base points \(\mathcal{P}\) are proper and let \(I\) denote its base ideal. Let \(J: = I(\mathcal{P};\mu)\) denote the associated ideal of fat points. a) The minimal graded free resolution of \(J\) is of the form \[ 0 \rightarrow R(-(d+2))^{d-2} \oplus R(-(d+1))^{s} \rightarrow R(-(d+1))^{d-4+s} \oplus R(-d)^{3} \rightarrow J \rightarrow 0, \] where \(s\) is the number of independent linear syzygies of the ideal \(I\). Furthermore, \(0\leq s \leq 1\), while \(s=1\) if and only if \(\mathfrak{F}\) is a de Jonquiéres map. b) The linear system \(J_{d+1}\) defines a birational mapping of \(\mathbb{P}^{2}\) onto the image in \(\mathbb{P}^{d+4+s}\). Moreover, in Section 3 the authors present a partial classification of homaloidal types according to the highest virtual multiplicity. Some classical inequalities for Cremona map stem from the consideration of the three highest virtual multiplicities, but not much has been obtained by stressing the role of the behavior of the map in the neighborhood of a single point with highest virtual multiplicity \(\mu_{1} = \mu\), so the authors study this case in detail. For instance, they show that for \(\mu=d-2\) any Cremona map on general points of such a homaloidal type is saturated (Proposition 3.1). In the last section the authors study the following question. Question. Let \((d, \mu_{1}, \dots)\) denote a proper homaloidal type and let \(F\) be a Cremona map of this type and general base points. If the base ideal of \(F\) is not saturated, then \(\mu_{1} \leq \lfloor d/2 \rfloor\). As a result the authors show that the base ideal \(I\) of a Cremona map \(\mathfrak{F} : \mathbb{P}^{2} \dashrightarrow \mathbb{P}^{2}\) of degree \(d\), whose virtual multiplicities are all even, is not saturated under the hypothesis that there are three virtual multiplicities equal to \(\lfloor d/2 \rfloor\). Cremona maps; homaloidal nets; fat points; syzygies Ramos, Z.; Simis, A., Homaloidal nets and ideals of fat points I, LMS J. Comput. Math., 19, 54-77, (2016) Syzygies, resolutions, complexes and commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Multiplicity theory and related topics, Rational and birational maps, Birational automorphisms, Cremona group and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Graded rings, Cohen-Macaulay modules, Divisors, linear systems, invertible sheaves Homaloidal nets and ideals of fat points. I

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(K\) be an algebraic function field over the field \(k\) and \(V\) the valuation ring of \(K\). Zariski proved the local uniformization problem in characteristic \(0\), i.e to find a regular local ring \(R\) essentially of finite type over \(k\) with quotient field \(K\) such that the valuation ring \(V\) dominates \(R\). He used uniformizing a singular algebraic hypersurface along the rank 1 valuation. He proved that the hypersurface can be uniformized by special Cremona transformations, the so--called Perron transformations. A generalization to valuations of arbitrary rank of the Perron transformation is given. This is the basis to prove local uniformization of a singular variety in characteristic \(0\) along a valuation of arbitrary rank. arbitrary rank valuation; local uniformization; Perron transformation Elhitti, S.: Perron transforms. Comm. algebra 42, 2003-2045 (2014) Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Singularities of curves, local rings Perron transforms

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system From the introduction: ``Linearization of line bundles in the presence of algebraic group actions is a basic notion of geometric invariant theory; it also has applications to the local properties of such actions. For example, given an action of a connected linear algebraic group \(G\) on a normal variety \(X\) over a field \(k\), and a line bundle \(L\) on \(X\), some positive power \(L^{\otimes n}\) admits a \(G\)-linearization (as shown by \textit{D. Mumford} et al. [Geometric invariant theory. 3rd enl. ed. Berlin: Springer-Verlag (1993; Zbl 0797.14004), Corollary 1.6] when \(X\) is proper, and by \textit{H. Sumihiro} [J. Math. Kyoto Univ. 15, 573--605 (1975; Zbl 0331.14008), Theorem. 1.6] in a more general setting of group schemes; when \(k\) is algebraically closed of characteristic \(0\), we may take for \(n\) the order of the Picard group of \(G\) as shown by \textit{F. Knop} et al. [in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63--75 (1989; Zbl 0722.14032)]). It can be inferred that \(X\) is covered by \(G\)-stable Zariski open subsets \(U_i\), equivariantly isomorphic to \(G\)-stable subvarieties of projectivizations of finite-dimensional \(G\)-modules; if \(G\) is a split torus, then the \(U_i\) may be taken affine (see [\textit{H. Sumihiro}, J. Math. Kyoto Univ. 14, 1--28 (1974; Zbl 0277.14008), Corollary 2], [\textit{H. Sumihiro}, J. Math. Kyoto Univ. 15, 573--605 (1975; Zbl 0331.14008), Theorem. 3.8, Corollary 3.11], [\textit{F. Knop} et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63--75 (1989; Zbl 0722.14032), Theorem 1.1]). In this article, we show that the above results on linearization of line bundles and the local properties of algebraic group actions hold under weaker assumptions than normality, if the Zariski topology is replaced with the étale topology. For simplicity, we state our main result in the case where \(k\) is algebraically closed: Theorem 1.1. Let \(X\) be a variety equipped with an action of a connected linear algebraic group \(G\). {\parindent=0.7cm\begin{itemize}\item[(i)] If \(X\) is seminormal, then there exists a torsor \(\pi : Y \to X\) under the character group of \(G\), and a positive integer \(n\) (depending only on \(G\)) such that \(\pi^*(L^{\otimes n})\) is \(G\)-linearizable for any line bundle \(L\) on \(X\).\item[(ii)] If in addition \(X\) is quasi-projective, then it admits an equivariant étale covering by \(G\)-stable subvarieties of projectivizations of finite-dimensional \(G\)-modules.\item[(iii)] If \(G\) is a torus and \(X\) is quasi-projective, then \(X\) admits an equivariant étale covering by affine varieties. \end{itemize}} In Section 2, we gather preliminary results on the Picard group of linear algebraic groups, and on the equivariant Picard group \(\mathrm{Pic}^G(X)\) which classifies \(G\)-linearized line bundles on a \(G\)-scheme \(X\); these results are variants of those in [\textit{H. Sumihiro}, J. Math. Kyoto Univ. 15, 573--605 (1975; Zbl 0331.14008); \textit{F. Knop} et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 77--87 (1989; Zbl 0705.14005); \textit{F. Knop} et al., in: Algebraische Transformationsgruppen und Invariantentheorie, DMV Semin. 13, 63--75 (1989; Zbl 0722.14032)]. In particular, when \(X\) is reduced, we obtain an exact sequence \[ \mathrm{Pic}^G(X){\varphi}{\longrightarrow}\mathrm{Pic}(X){\psi}{\longrightarrow}\mathrm{Pic}(G \times X)/p_2^*\mathrm{Pic}(X), \] where \(\varphi\) denotes the forgetful map, and the obstruction map \(\psi\) arises from the pull-back under the action morphism \(G \times X \to X\) (see Proposition 2.10). Finally, we show that the obstruction group \(\mathrm{Pic}(G \times X)/p_2^* \mathrm{Pic}(X)\) is \(n\)-torsion if \(X\) is normal, where \(n\) is a positive integer depending only on \(G\) (Theorem 2.14). The obstruction group is studied further in Section 3. We construct an injective map \(c : H^1_{et}(X,\hat{G}) \to \mathrm{Pic}(G \times X)/p_2^*\mathrm{Pic}(X)\), where the left-hand side denotes the first étale cohomology group with coefficients in the character group of \(G\) (viewed as an étale sheaf); recall that this cohomology group classifies \(\hat{G}\)-torsors over \(X\). In Section 4, we present several applications of our analysis of the obstruction group. We first show that linearizability is preserved under algebraic equivalence (Proposition 4.1). Then we obtain a version of Theorem 1.1 over an arbitrary base field (Theorems 4.4, 4.7 and 4.8). Finally, we show that the seminormality assumption in Theorem 1.1 (i) and (ii) may be suppressed in prime characteristics (Subsection 4.3).'' line bundle; linearization; seminormal Brion, M.\!, On linearization of line bundles, J. Math. Sci. Univ. Tokyo, 22, 113-147, (2015) Divisors, linear systems, invertible sheaves, Linear algebraic groups over arbitrary fields, Picard groups, Geometric invariant theory, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On linearization of line bundles

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Consider an \((n-s)\)-dimensional algebraic variety \(W\) defined over an infinite field \(k\) of nonzero characteristic \(p\) and irreducible over this field. Let \(W\) be a subvariety of the projective space of dimension \(n\). We prove that the local ring of \(W\) has a sequence of local parameters represented by \(s\) nonhomogeneous polynomials with the product of degrees less than the degree of the variety multiplied by a constant depending on \(n\). This allows us to prove the existence of effective smooth cover and smooth stratification of an algebraic variety in the case of ground field of nonzero characteristic. The paper extends the analogous results of the author obtained earlier in the case of zero characteristic of the ground field. A. L. Chistov, Efficient construction of local parameters of irreducible components of an algebraic variety in nonzero characteristic, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 (2005), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13, 248 -- 278, 284 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 3, 480 -- 496. Computational aspects of higher-dimensional varieties Efficient construction of local parameters of irreducible components of an algebraic variety in nonzero characteristic

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system These notes summarize part of my research work as a SAGA post-doctoral fellow. We study a class of polynomial interpolation problems which consists of determining the dimension of the vector space of homogeneous or multi-homogeneous polynomials vanishing together with their partial derivatives at a finite set of general points. After translating the problem into the setting of linear systems in projective spaces or products of projective lines, we employ algebro-geometric techniques such as blowing-up and degenerations to calculate the dimension of such vector spaces. We compute the dimensions of linear systems with general points of any multiplicity in \(\mathbb P^n\) in a family of cases for which the base locus is only linear ([\textit{M. C. Brambilla} et al., Trans. Am. Math. Soc. 367, No. 8, 5447--5473 (2015; Zbl 1331.14007)]). Moreover we completely classify linear systems with double points in general position in products of projective lines \((\mathbb P^1)^n\) ([\textit{A. Laface} and \textit{E. Postinghel}, Math. Ann. 356, No. 4, 1455--1470 (2013; Zbl 1275.14041)]) and we relate this to the study of secant varieties of Segre-Veronese varieties. homogeneous or multi-homogeneous polynomials; algebro-geometric techniques; linear systems Interpolation in approximation theory, Projective techniques in algebraic geometry Polynomial interpolation problems in projective spaces and products of projective lines

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R=K[X_{ij}]/I_ p\) be the coordinate ring of a determinantal variety. Here K is a field of characteristic zero, \(X_{ij}\) are the coordinate functions on the affine space \(M_{m\times n}(K)\) of all \(m\times n\) matrices over K and \(I_ p\) with \(1\leq p\leq \min (m,n)\) is the ideal generated by the \(p\times p\) minors of \(X=(X_{ij})\). The problem is to find explicitly a minimal free resolution of R over \(K[X_{ij}]\). The first approach to this question is due to Lascoux, who gave an explicit description of all components of a minimal resolution of R for any m, n, p. But his proofs seem to be incomplete. Here the authors present a somewhat different approach to this problem. Their methods are influenced by a construction of \textit{T. Gulliksen} and \textit{N. Negard} [C. R. Acad. Sci., Paris, Sér. A 274, 16-18 (1972; Zbl 0238.13015)], and they are mainly based on two papers of Akin, Buchsbaum and Weyman, using the technique of Schur complexes [see \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman}, Adv. Math. 44, 207-278 (1982; Zbl 0497.15020) and 39, 1-30 (1981; Zbl 0474.14035)]. In a suitable generalization of the fundamental idea in the paper cited at last they can show that all modules of Lascoux's resolution can be obtained as the components of the homology complex coming from a certain double complex \({\mathcal L}^ k_{\bullet \bullet}\) of complexes (for \({\mathcal L}^ k_{\bullet \bullet}\) see theorem 1.8). The authors mention that the same procedure doesn't work in general, i.e for arbitrary characteristic. determinantal variety; minimal free resolution P. Pracacz and J. Weyman, Complexes associated with trace and evaluation: Another approach to Lascoux's resolution, Adv. Math. 57 (1985), 163--207. Determinantal varieties, Global theory and resolution of singularities (algebro-geometric aspects) Complexes associated with trace and evaluation. Another approach to Lascoux's resolution

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A rank one local system on the complement of a hyperplane arrangement is said to be admissible if it satisfies certain non-positivity condition at every resonant edges. It is known that the cohomology of admissible local system can be computed combinatorially. In this paper, we study the structure of the set of all non-admissible local systems in the character torus. We prove that the set of non-admissible local systems forms a union of subtori. The relations with characteristic varieties are also discussed. line arrangements; admissible local systems; characteristic variety S. Nazir, M. Torielli, M. Yoshinaga, On the admissibility of certain local systems. Topology Appl. 178 (2014), 288-299.DOI: 10.1016/j.topol.2014.10.001 (Co)homology theory in algebraic geometry, Configurations and arrangements of linear subspaces, Relations with arrangements of hyperplanes, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) On the admissibility of certain local systems

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \({\mathcal A}\) be a spanning subset of \(\mathbb{Z}^{n+1}\) consisting of \(r\) elements, and let \(\alpha\in \mathbb{C}^{n+1}\). In the late eighties Gel'fand, Kapranov and Zelevinskij associated with \({\mathcal A}\) and \(\alpha\) a holonomic system of differential equations in \(\mathbb{C}^r\), called the \({\mathcal A}\)-hypergeometric system with exponent (or parameter) \(\alpha\). Its solutions are called the \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) [see \textit{I. M. Gel'fand}, \textit{A. V. Zelevinskij} and \textit{M. M. Kapranov}, Funct. Anal. Appl. 23, No. 2, 94-106 (1989; Zbl 0721.33006); Adv. Math. 84, No. 2, 255-271 (1990; Zbl 0741.33011)]. In the literature \({\mathcal A}\)-hypergeometric systems are also called GKZ-systems. The paper under review studies the case of \({\mathcal A}\)-hypergeometric systems associated with monomial curves, which corresponds to the case \(n=1\). All rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) are shown to be Laurent polynomials. This property is proven by counterexample not to be true in the general case \(n>1\). The rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\in \mathbb{Z}^2\) are shown to span a space of dimension at most 2. The value 2 is attained if and only if the monomial curve is not arithmetically Cohen-Macaulay. For all values of \(\alpha\), the holonomic rank \(r(\alpha)\) of the system is proven to satisfy the inequalities \(d\leq r(\alpha)\leq d+1\). Moreover \(r(\alpha)= d+1\) exactly for all \(\alpha\in \mathbb{Z}^2\) for which the space of rational solutions has dimension 2. The inequalities for the holonomic rank have also been obtained using different methods by \textit{M. Saito}, \textit{B. Sturmfels} and \textit{N. Takayama} [Gröbner deformations of hypergeometric differential equations. Springer-Verlag (2000; Zbl 0946.13021)]. GKZ-systems; \({\mathcal A}\)-hypergeometric function; \({\mathcal A}\)-hypergeometric system Eduardo Cattani, Carlos D'Andrea, and Alicia Dickenstein, The \?-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179 -- 207. Other hypergeometric functions and integrals in several variables, Families, fibrations in algebraic geometry, Deformations of analytic structures, Basic hypergeometric functions in one variable, \({}_r\phi_s\) The \({\mathcal A}\)-hypergeometric system associated with a monomial curve

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Fix an algebraically closed field \(k\) of arbitrary characteristic, and a sequence \(p=(p_ 0,\dots,p_ n)\) of integers with \(p_ i\geq 1\). \(G(p)\) denotes the set of \((n+1)\times (n+1)\) diagonal invertible matrices over \(k\) which act on \(k^{n+1}\) by multiplication. The coordinate ring \(S\) of \(k^{n+1}\) is thus graded by the character group \(L_{(p)}\). To a sequence of pairwise distinct \((\lambda_ 0=\infty, \lambda_ 1=0, \lambda_ 2=1,\dots,\lambda_ k)\) of \(P_ 1(k)\) is attached the homogeneous ideal of \(S\) generated by the elements \(f_ i=X_ i^{p_ i}-x_ 1^{p_ 1}-\lambda_ iX_ 0^{p_ 0}\). \(G(p)\) acts on \(k^{n+1}-(0)\) and on the subvariety determined by the zeros of the \(f\). The respective quotients are denoted \(P_ n(p)\) and \(C(p,\lambda)\). These are endowed with structure sheaves by extending the presheaf which assigns the (graded) ring \(S_ f\) to the non-zeros of homogeneous \(f\) in \(S\). The resulting structures are termed weighted projective space and weighted projective line. Let \(X\) stand for either object. The authors prove that the category of (graded) coherent sheaves on \(X\) is equivalent to the category of positively graded modules over the homogeneous coordinate ring of \(X\) which admit no simple submodules or extensions by such. They further demonstrate that the category of graded coherent sheaves, and its subcategory of vector bundles, have many properties analogous to the same objects for ordinary projective space. - For the case \(C=C(p,\lambda)\), the authors define a tilting sheaf to be a graded coherent sheaf \(T\) with \(\text{Ext}^1(T,T)=0\), and \(T\) a generator of the derived category of bounded complexes of graded coherent sheaves on \(C\) (the latter is denoted \(D^b(\text{coh}(C))\). They prove that \(D^b(\text{coh}(C))\) is equivalent to the derived category of bounded complexes of finitely generated \(\text{End}(T)\)-modules. Finally, they establish that the canonical algebras of Ringel arise as endomorphism rings of tilting sheaves. Thus, the results on categories of sheaves over graded projective lines translate to theorems about modules over canonical algebras. [For the entire collection see Zbl 0619.00007.] weighted projective space; weighted projective line; graded coherent sheaves; tilting sheaf; derived category of bounded complexes W. Geigle and H. Lenzing, \textit{A class of weighted projective curves arising in the representation theory of finite dimensional algebras}, in \textit{Lectures Notes in Mathematics. Vol. 1273: Singularities, Representation of Algebras, and Vector Bundles}, Springer, Berlin Germany (1987). Special algebraic curves and curves of low genus, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) A class of weighted projective curves arising in representation theory of finite dimensional algebras

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0744.00034.] The article under review is devoted to a theory of linear systems over singular K3-surfaces (i.e. of normal projective algebraic surfaces, such that the minimal resolution is K3 in the usual sense). This is motivated by relations with the classification theory for Fano threefolds. -- The author studies trees of curves arising in the following way: First of all, let \(H\) be an effective divisor on the smooth K3-surface \(X\), \(| H |=| C |+\Delta\) \((\Delta\) the fixed part of the linear system), and consider the condition: \((*)\) One of the following assertions is satisfied: (i) \(C^ 2>0\), \(| C |\) contains an irreducible curve and has no fixed points, (ii) \(C^ 2=0\), \(| C |=m \cdot | E |\), \(| E |\) an elliptic pencil, (iii) \(C=0\). Then, for \(C\) and \(\Delta\) satisfying \((*)\), the condition \(| C+\Delta |=| C |+\Delta\) is shown to be equivalent with the property: \(G(C,\Delta)\) (:=dual graph of the intersections of the irreducible components) is a tree, and it has no subtrees \(D^ \sim_ m\), \(D^ \sim_ m(C)\) \((m \geq 4)\), \(E^ \sim_ m\), \(E^ \sim_ m(C)\) \((m=6,7,8)\), \(B^ \sim_ m(C)\) \((m \geq 2)\), \(G^ \sim_ 2(C)\), where \(C\) denotes the terminal element in one ``branch'' of maximal length in the corresponding graph. This theorem reduces the description of all possible graphs \(G(C,\Delta)\) to the description of trees containing at most one curve \(C\) with \(C^ 2 \geq 0\) and with all other irreducible components nonsingular rational curves. -- Now let \({\mathcal T}\) be such a tree, \(G(C)\) its intersection graph, \(G\) the intersection graph of the smooth rational components of \({\mathcal T}\). The Hodge index theorem allows at most one positive square for the corresponding intersection matrix. Thus a classification is obtained into hyperbolic, parabolic and elliptic type for \(G\). What follows is a closer investigation of all possible trees \(G(C)\) (the rank of the Picard lattice is \(\leq 22\), resp. \(\leq 20\) for the case of characteristic 0), reducing the problem to questions of combinatorics and linear algebra. The general case is treated in this way: Let \(\sigma:X \to Y\) be the minimal resolution of a singular K3-surface \(Y\), \(\Delta_ s:=\sum b_ jF_ j\) \((b_ j \geq 0)\), and \(F_ j\) the components of the exceptional divisor on \(X\). Further, let \(D\) be an effective divisor on \(X\). A complete Weil linear system is the image \(\overline D=\sigma_ *(| D |+\Delta_ s)\), which is stabilizing for large \((b_ j)\). As before, a graph \(G(C,\Delta)\) can be defined, and there is an analog of the condition \((*)\) with \(\Delta=\Delta_ r+\Delta_ s\), \(\Delta_ r=\sum a_ i \Gamma_ i\) \((a_ i \in \mathbb{N})\) fixed, and \(\Delta_ s=\sum b_ jF_ j\), \((b_ j>>0)\), giving rise to a similar characterization of the equality \(| C+\Delta |=| C |+\Delta\). -- For the case of \(\text{rk(Pic} Y)=1\), a more detailed discussion is included. linear systems over singular K3-surfaces; Fano threefolds; effective divisor; Hodge index theorem; complete Weil linear system Nikulin, V.V.: Weil linear systems on singular \(K3\) surfaces. In: Algebraic Geometry and Analytic Geometry (Tokyo, 1990). ICM-90 Satellite Conference Proceedings, pp. 138-164. Springer, Tokyo (1991) \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves Weil linear systems on singular K3 surfaces

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(R\) be a regular local ring with maximal ideal \(\mathcal{M}\) and \(\hat R\) its \(\mathcal{M}\)-adic completion. Then \(R\subset \hat R\). An element \(f\in R\) can be expanded as a finite sum \(\displaystyle f=\sum_{\alpha\in\mathbb{Z}^n_{\geq 0}}a_{\alpha}x^{\alpha}\), where \(a_{\alpha}\in U(R)\cup\{0\}\), \(x=(x_1,\ldots,x_n)\) is a regular system of parameters of \(R\), \(\alpha=(\alpha_1,\ldots,\alpha_n)\) and \(x^{\alpha}=x_1^{\alpha_1}\ldots x_n^{\alpha_n}\). The main results of this paper are factorization theorems for elements in \(R\) and polynomials with coefficients in \(R\), using convex geometry. We need some preliminaries to formulate the first theorem. For \(\displaystyle f=\sum_{\alpha\in\mathbb{Z}^n_{\geq 0}}a_{\alpha}x^{\alpha}\in R\) a nonzero element, the Newton polyhedron \(\Delta(f)\) of \(f\) is the convex hull of the set \(\{\alpha;~a_{\alpha}\not=0\}+\mathbb{R}^n_{\geq 0}\). A set \(\Delta\subset\mathbb{R}^n_{\geq 0}\) is called a Newton polyhedron if \(\Delta=\Delta(f)\) for some \(0\not=f\in R\). For \(\xi\in\mathbb{R}^n_{\geq 0}\) and \(\Delta\subset\mathbb{R}^n_{\geq 0}\) a Newton polyhedron, we call the set \(\Delta^{\xi}=\{a\in \Delta;~<\xi,a>=min_{b\in\Delta}<\xi,b>\}\) a face of \(\Delta\). Here \(<.,.>\) denotes the standard scalar product. A face \(\Delta^{\xi}\) is compact if and only if \(\xi\in\mathbb{R}^n_>\). A face of dimension \(1\) is called an adge and a compact edge of a Newton polyhedron is called a loose adge if it is not contained in any compact face of dimension \(\geq 2\). Now, we are ready to give the first main result of the paper. Let \(R\) be a regular local ring and \(0\not=f\in R\). Assume that the Newton polyhedron \(\Delta(f)\) has a loose edge \(E\). If \(f_{/E}\) is a product of two relatively prime polynomials \(G\) and \(H\), where \(G\) is not divided by any variable, then there exists \(g,h\in\hat R\) such that \(f=gh\) and \(g_{/E_1}=G\), \(h_{/E_2}=H\) for some \(E_1\) and \(E_2\) such that \(E\) is the Minkowski sum of \(E_1\) and \(E_2\), \(E=E_1+E_2\). irreducibility; formal power series; Newton polyhedron Formal power series rings, Polynomials in general fields (irreducibility, etc.), Singularities in algebraic geometry Loose edges and factorization theorems

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(A\) be a regular local ring containing \(1/2\), which is either equicharacteristic, or is smooth over a d.v.r. of mixed characteristic. We prove that the product maps on derived Grothendieck-Witt groups of \(A\) satisfy the following property: given two elements with supports which do not intersect properly, their product vanishes. This gives an analogue for ``oriented intersection multiplicities'' of Serre's vanishing result for intersection multiplicities. It also suggests a vanishing conjecture for arbitrary regular local rings containing \(1/2\), which is analogous to Serre's (which was proved independently by Roberts, and Gillet and Soulé). J. Fasel and V. Srinivas, A vanishing theorem for oriented intersection multiplicities, Math. Res. Lett., 15(3) (2008), 447--458. Regular local rings, Vanishing theorems in algebraic geometry A vanishing theorem for oriented intersection multiplicities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The author presents some properties over rings which remain unchanged if one considers non-reduced rings \(R\) or the corresponding reduced ring \(\overline R\) (modulo the nil radical). Among others the following results are obtained: (1) For locally nilpotent derivations \(D\) having a slice the cancellation problem (i.e. \(D\) is conjugate to \(\partial_{X_1}\) over \(R[X_1,\dots, X_n]\)) is equivalent to that over \(\overline R[X_1,\dots, X_n]\); (2) if \(F\in R[X_1,\dots, X_n]^n\), \(F^s= (X_1,\dots, X_n)\) for some \(s\geq 1\), the linearisation conjecture over \(R\) (i.e. \(F\) is linearizable by conjugation) is equivalent to that over \(\overline R\). non-reduced rings; locally nilpotent derivation; cancellation problem; linearization conjecture; reduced rings S. Maubach: The linearisation conjecture and other problems over nonreduced rings , Comm. Algebra 30 (2002), 1693--1704. Derivations and commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) The linearisation conjecture and other problems over nonreduced rings.

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A classic result by \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)] says that the notion of an (infinite-dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for \(n=0)\) of the locality of more general notions of quasi-coherent sheaves related to (infinite-dimensional) \(n\)-tilting modules and classes. Here, we prove the latter locality for all \(n\) and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is Noetherian). tilting module; locally tilting quasi-coherent sheaf; Zariski locality; Mittag-Leffler module Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, General module theory in associative algebras, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Free, projective, and flat modules and ideals in associative algebras, Other special types of modules and ideals in commutative rings, Homological functors on modules of commutative rings (Tor, Ext, etc.), Abelian categories, Grothendieck categories Zariski locality of quasi-coherent sheaves associated with tilting

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \((R,\mathfrak{m})\) denote a local ring. For two ideals \(I, J \subset R\) denote by \(I : \langle J\rangle = \bigcup_{m \geq 1} I : J^m.\) Very often it is of some interest whether the family of ideals \(\{I^n :\langle J\rangle\}_{n \geq 1}\) is equivalent to the \(I\)-adic topology, i.e., for each \(n \geq 1\) there is an \(m(n) \geq 1\) such that \(I^{m(n)}:\langle J\rangle \subseteq I^n.\) The situation when \(m(n) = n +k\) for a certain integer \(k \geq 1\) was characterized by the reviewer [Math. Nachr. 129, 123--148 (1986; Zbl 0606.13001)]. He posed the problem whether the equivalence of \(\{I^n :\langle J\rangle\}_{n \geq 1}\) to the \(I\)-adic topology implies that one might choose \(m(n) = nk\) for a certain integer \(k \geq 1,\) now called linear equivalence. This problem was solved by \textit{I. Swanson} [Math. Z. 234, No. 4, 755--775 (2000; Zbl 1010.13015)]. For symbolic powers of a prime ideal \(\mathfrak{p} \subset R\) this implies that \(\mathfrak{p}^{(nk)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1.\) While a priori \(k\) depends on the ideal \(\mathfrak{p},\) in their paper \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] have shown that \(\mathfrak{p}^{(nd)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1\) and \(\dim R = d\) for a prime ideal \(\mathfrak{p}\) of a regular local ring \(R.\) In the paper under review the authors study when there is a uniform \(k \geq 1\) such that \(I^{nk}:\langle J\rangle \subseteq I^n\) for all \(n \geq 1\) and all ideals \(I,\) provided \(\{I^n :\langle J\rangle\}_{n \geq1}\) is equivalent to the \(I\)-adic topology. As the main result of the paper the authors prove the following Theorem. Let \(R\) be an equicharacteristic, local domain such that \(R\) is an isolated singularity. Assume that \(R\) is either essentially of finite type over a field of characteristic zero or \(R\) has positive characteristic and is \(F\)-finite. Then there exists an integer \(k \geq 1\) with the property that for all ideal \(I \subset R\) such that the symbolic topology of \(I\) is equivalent to the \(I\)-adic topology, \(I^{(nk)} \subseteq I^n\) for all \(n \geq 1.\) The main ingredients of the proof of their result is threefold: (1) The relation between the jacobian ideal and symbolic powers as established by Hochster and Huneke [loc. cit.]. (2) The uniform Artin-Rees Theorem as it was shown by the first author [Invent. Math. 107, No. 1, 203--223 (1992; Zbl 0756.13001)]. (3) A uniform Chevalley Theorem proved in the paper: Let \(R\) be an analytically unramified local ring. Then there is an integer \(k \geq 1\) with the property: For all ideals \(I \subset R\) and all multiplicatively closed subsets \(S\) in \(R\) such that the topology \(\{ I^n R_S \cap R\}_{n \geq 1}\) is finer then the \(\mathfrak{m}\)-adic topology it follows that \(I^{nk}R_S \cap R \subset \mathfrak{m}^n\) for all \(n \geq 1.\) equivalence of ideal topologies; uniform equivalence; uniform Artin-Rees Lemma; Chevalley's Theorem Huneke, C.; Katz, D.; Validashti, J., Uniform equivalence of symbolic and adic topologies, Ill. J. Math., 53, 325-338, (2009) Ideals and multiplicative ideal theory in commutative rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects and applications of commutative rings, Effectivity, complexity and computational aspects of algebraic geometry Uniform equivalence of symbolic and adic topologies

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The key point in the resolution of singularities is the choice of the centers for the blowing-ups. In characteristic zero the center as a subset of the locus of highest multiciplity is determined by induction passing to a so-called hypersurface of maximal contact (eliminating one variable). In characteristic \(p>0\) this is not possible in general. The multiplicity of an embedded hypersurface at a point can be described in terms of differential operators as well as in terms of general projections defined at étale neighbourhoods of this point. Both approaches are related in the paper. Invariants of embedded hypersurfaces are studied in terms of differential operators, which express properties of the ramification of the morphism. A central result in multiplicity theory of hypersurfaces is a form elimination of one variable in the description of highest multiplicity locus. In characteristic \(0\) this form of elimination is achieved with the notion of Tschirnhausen polynomials introduced by Abhyankar. This is a key point in the proof of embedded desingularization. A characteristic free approach to this form of elimination is given. Given a \((d-1)\)-dimensional hypersurface in a smooth \(d\)-dimensional scheme. The approach based on projections on smooth \((d-1)\)-dimensional schemes. The behaviour of invariants related to this form of elimination is discussed. Villamayor, O.: Hypersurface singularities in positive characteristic. Adv. in Math. 213 (2007), no. 2, 687-733. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties Hypersurface singularities in positive characteristic

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Definition. A linear system \(g^r_n\) on \(C\) is very special if \(r\geq 1\) and \(\dim |K_C- g^r_n |\geq 1\). (Here \(K_C\) is a canonical divisor on \(C\).) A base point free complete very special linear system \(g^r_n\) on \(C\) is trivial if there exists an integer \(m\geq 0\) and an effective divisor \(E\) on \(C\) of degree \(md-n\) such that \(g^r_n= |mg^2_d-D |\) and \(r= {m^2+3m \over 2}- (md-n)\). A complete very special linear system \(g^r_n\) on \(C\) is trivial if its associated base point free linear system is trivial. By non-trivial linear systems we mean complete very special linear systems which are not trivial. In this paper we consider the following question. For \(n\in \mathbb{Z}_{\geq 1}\) find \(r(n)\) such that there exists a nontrivial linear system \(g_n^{r(n)}\) on \(C\) but no such linear system \(g_n^{r(n)+1}\). Our main result is the following theorem: Let \(g^r_n\) be a base point free non-trivial linear system on \(C\). Write \(r= (x+1) (x+2)/2 -\beta\) with \(x,\beta\) integers satisfying \(x\geq 1\), \(0\leq \beta\leq x\). Then \(n\geq n(r): =(d-3) (x+3)- \beta\). very special linear system Coppens M., Kato T.: Non-trivial linear systems on smooth plane curves. Math. Nachr. 166, 71--82 (1994) Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus Non-trivial linear systems on smooth plane curves

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The paper is concerned with normal two dimensional singularities (V,p), in particular with elliptic singularities in the sense of Wagreich \((p_ a=1)\). The goal is, to compare Zariski's canonical resolution (successive blowing up points and normalization) with the minimal resolution of (V,p). One main result is the following: Let (V,p) be elliptic and Gorenstein and let E be the minimal elliptic cycle on the minimal resolution of (V,p). - \((i):\quad (V,p)\) is absolutely isolated (i.e. no normalization occurs in Zariski's canonical resolution) if and only if \(E^ 2\leq -3\). - \((ii):\quad Zariski's\) canonical resolution gives the minimal resolution if and only if \(E^ 2\leq -2\). - This extends known result of H. B. Laufer and S. S.-T. Yau. Moreover the author gives a precise description of the relation between Zariski's canonical resolution and the minimal resolution without any assumption on \(E^ 2.\) In the proofs the author uses a formula for the geometric genus \(p_ g\) of a normal 2-dimensional hypersurface singularity in terms of resolution data obtained by successive blowing up smooth centers. This formula is proved in the first part of the paper. normal two dimensional singularities; elliptic singularities; Zariski's canonical resolution; minimal resolution M. Tomari, A \(p_g\)-formula and elliptic singularities, Publ. Res. Inst. Math. Sci. 21 (1985), no. 2, 297--354. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry A \(p_ g\)-formula and elliptic singularities

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system This paper contains two parts. In the first one, the author gives an example of a Stanley-Reisner ring \(k[\Delta]\) whose defining ideal is generated by monomials of degree at most 2 and such that the maps \[ \to F_{i+1}\to F_ i\to F_{i-1}\to\cdots\to F_ 0\to k[\Delta]\to 0, \] in a minimal free resolution of \(k[\Delta]\) as a module over a polynomial algebra \(A\), are not represented by matrices with all components of degree at most 2. (A counterexample to a question of Watanabe). In the second part the author proves that the canonical map \(\hbox{Ext}^ i_ A(k,k[\Delta])\to H^ i_ m(k[\Delta])\) corresponds to the map induced by the inclusion map of certain subcomplexes of \(\Delta\). local cohomology; Stanley-Reisner ring Miyazaki, Mitsuhiro, On the canonical map to the local cohomology of a Stanley-Reisner ring, Bull. Kyoto Univ. Educ., Ser. B, 79, 1-8, (1991) Simplicial sets and complexes in algebraic topology, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Local cohomology and algebraic geometry On the canonical map to the local cohomology of a Stanley-Reisner ring

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system The goal of this paper is to present the theoretical basis for an algorithmic implementation of embedded resolution of arithmetical schemes. To this end, the authors use as a starting point \textit{V. Cossart} et al.'s paper [Desingularization: invariants and strategy. Application to dimension 2. With contributions by Bernd Schober. Cham: Springer (2020; Zbl 1477.14003)] on resolution of two-dimensional schemes. The first step is to construct an upper semi-continuous function that stratifies the singular locus of the variety in a way that the maximum value determines the centers to blow up. In \textit{V. Cossart} et al.'s paper [Desingularization: invariants and strategy. Application to dimension 2. With contributions by Bernd Schober. Cham: Springer (2020; Zbl 1477.14003)] the upper-semi-continuous function is constructed taking into account the Hilbert-Samuel function. However, this function is difficult to implement. Thus the authors use a slightly different strategy following \textit{S. Encinas} and \textit{O. Villamayor}'s approach in [Rev. Mat. Iberoam. 19, No. 2, 339--353 (2003; Zbl 1073.14021)]. There, resolution of algebraic varieties is proved using the order of an ideal as main invariant. However, to use this approach, since the goal is to work with aritmetical schemes, some work is done to show that a suitable stratification of the singular locus can be constructed using the order of an ideal as main invariant. Next, there is another non-trivial problem to face. How does one \textit{compute} the set of points with a given order of an ideal where there is not a good theory of differential operators at disposal? When working over arbitrary fields, this still can be worked out (e.g. if there are \(p\)-basis). But, If there is no base field, and the base is a Dedeking domain, then a technique is developed to search for the so called \textit{wbad primes}, i.e., primes over wchich singularities can be found (and that cannot be described using differential operators). The paper is very nicely written and several examples are presented to clarify some difficult points. resolution of singularities; differential operators Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Computational aspects and applications of commutative rings, Software, source code, etc. for problems pertaining to commutative algebra Embedded desingularization for arithmetic surfaces -- toward a parallel implementation

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Given an algorithm for resolution of singularities that satisfies certain conditions (``a good algorithm''), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme \(T\)) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced \(T\), this parameter scheme can be naturally expressed as a disjoint union of locally closed sets \(T_j\), such that the induced family on each part \(T_j\) is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families. resolution of singularities; algorithmic resolution; simultaneous resolution; Hilbert schemes Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. Global theory and resolution of singularities (algebro-geometric aspects) On algorithmic equi-resolution and stratification of Hilbert schemes

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system In the paper under review, the authors study the fine structure of the linearly presented perfect ideals of codimension two in three variables with particular emphasis on the Cohen-Macaulay, normal and fiber type properties of the associated Rees algebra and the special fiber. There is a vast literature on codimension two perfect ideals in the case the ideal satisfies the usual generic properties or the so-called ($G_d$) condition. One first attempt to get away from the ($G_d$) condition or others of similar nature is a recent result of \textit{N. P. H. Lan} [J. Pure Appl. Algebra 221, No. 9, 2180--2191 (2017; Zbl 1453.13018)]. In the present paper, the authors recover and extend his work. More precisely, the authors introduce the so-called \textit{chaos invariant} associated to the Hilbert-Burch matrix of $I$. In the case the chaos invariant is one (minimal possible), they recover Lan's result with additional contents. They also give applications of these results to three important models: linearly presented ideals of plane fat points, reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals. \par In the paper the authors strongly focus on the behavior of the ideals of minors of the corresponding Hilbert-Burch matrix and on conjugation features of the latter. linear presentation; Cohen-Macaulay ideal; special fiber; Rees algebra; reduction number Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Linkage, complete intersections and determinantal ideals, Syzygies, resolutions, complexes and commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Graded rings, Determinantal varieties Linearly presented perfect ideals of codimension 2 in three variables

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Let \(G\) be a connected reductive linear algebraic group and \(\rho\colon G\to\text{GL}(M)\) a rational representation of \(G\). In this paper the author studies the diagonal action of \(G\) on \(M^p\). An element of \(M^p\) is called `nilpotent' if zero is in the closure of its \(G\)-orbit; the set of nilpotent elements in \(M^p\) forms a \(G\)-stable closed cone called the `null-cone' in \(M^p\). Let \(c(M^p)\) denote the number of irreducible components of this null-cone. The main result of this paper is an algorithm to calculate \(\lim_{p\to\infty}c(M^p)\) given the weight system of \(M\). The fact that \(c(M^p)\) stabilizes at some finite value of \(p\) is shown by \textit{H. Kraft} and \textit{N. R. Wallach} [in Pac. J. Math. 224, No. 1, 119-139 (2006; Zbl 1124.20030)]; they also provide a characterization of the null-cone which is central to the construction of the algorithm. The author goes on to apply the algorithm to show that if \(G\) is semisimple and the characteristic is zero, only a few \(G\)-modules \(M\) have the property that \(c(M^p)\) is small for all \(p\). A conjecture is also presented concerning the question of when \(c(M^p)=1\) for all \(p\). null-cones of representations; reductive groups; connected reductive linear algebraic groups; rational representations; diagonal actions; nilpotent elements; numbers of irreducible components; algorithms Representation theory for linear algebraic groups, Geometric invariant theory, Group actions on varieties or schemes (quotients) Counting components of the null-cone on tuples.

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system [For the entire collection see Zbl 0632.00003.] The author gives a new proof for her result in Am. J. Math. 104, 545-552 (1982; Zbl 0496.13008) that the analytic spread \(\ell (mS)\) is at most d- 1 if (R,m)\(\varsubsetneq (S,n)\) is an extension of d-dimensional regular local rings having the same quotient field. The idea is to see, without assuming regularity of S, what consequences \(\ell (mS)=d\) might have. One is that \(R/m=S/n\); then the original theorem follows quickly by using this fact together with Zariski's main theorem. Another application is to show that if S is a local UFD birationally dominating a 2-dimensional regular local ring, then S itself is regular. birational domination; analytic spread; regular local rings; Zariski's main theorem; UFD Extension theory of commutative rings, Regular local rings, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Rational and birational maps Maximal analytic spread in birational extensions of regular local rings

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system A birational correspondence between nonsingular projective algebraic surfaces defined over an algebraically closed field can be factored into a product of quadratic transforms and inverses of quadratic transforms. This is an easy consequence of the following two theorems. Domination: If X and Y are birationally equivalent non-singular projective algebraic surfaces defined over an algebraically closed field, there exists a nonsingular projective algebraic surface Z such that Z is an iterated monoidal transform of Y with nonsingular irreducible centers and Z dominates X. Factorization: If X and Z are birationally equivalent nonsingular projective algebraic surfaces defined over an algebraically closed field and Z dominates X, then Z is an iterated monoidal transform of X with nonsingular irreducible centers. In dimension three domination remains true, but factorization is false. A local version of this problem is: Let R and S be three-dimensional regular local rings which are spots over an algebraically closed field k. Assume that R and S have the same quotient field K, that \(tr\deg_ kK=3\), and that there exists a valuation ring V of K/k such that V dominates R and V dominates S. Does there exist a regular local ring T such that T is an iterated monoidal transform (with nonsingular irreducible centers) of R along V and T is an iterated monoidal transform (with nonsingular irreducible centers) of S along V? In what follows it is proved that the answer to the above question is yes if \(restr\deg_ kV>0\) or if the rational rank of V \(=3\). These results are proved by constructing algorithms which explicitly give the required sequences of transformations. birational correspondence; domination; factorization; three-dimensional regular local rings Chris Christensen, Strong domination/weak factorization of three-dimensional regular local rings, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 21 -- 47 (1984). Chris Christensen, Strong domination/weak factorization of three-dimensional regular local rings. II, J. Indian Math. Soc. (N.S.) 47 (1983), no. 1-4, 241 -- 250 (1986). Regular local rings, Rational and birational maps, \(3\)-folds Strong domination/weak factorization of three dimensional regular local rings

Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system Computation of cohomology of local systems is a difficult task. In this paper it is proved that for all rank one local systems \(L\) on an 1-formal smooth complex algebraic variety \(M\), except finitely many local systems in any irreducible component \(W\) of the first characteristic variety, \(H^1(M,L)\) can be computed from the cohomology algebras \(H^*(M_W,\mathbb{C})\), where \(M_W=M\) if \(W\) is a non-translated component but could be a smaller open subset of \(M\) otherwise. This is applied to obtain a simple topological proof of the fact that if the dimension of \(H^1(M,L')\), as \(L'\) varies within \(W\) with \(\dim W>0\), jumps at \(L\in W\), then \(L\) is of finite order. This result also follows by applying the more difficult results of \textit{A. Dimca, S. Papadima} and \textit{A. I. Suciu} [Int. Math. Res. Not. 2008, Article ID rnm119 (2008; Zbl 1156.32018)], \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)], and \textit{N. Budur} [Adv. Math. 221, No. 1, 217--250 (2009; Zbl 1187.14024)]. As a corollary, if \(M\) is quasi-projective, then \(H^1(M,L)=H^1(M,L^{-1})\). local system; constructible sheaf; twisted cohomology; characteristic variety; resonance variety A. Dimca: On admissible rank one local systems, J. Algebra (2008), doi:10.1016/j.jalgebra.2008.01.039. Homotopy theory and fundamental groups in algebraic geometry On admissible rank one local systems