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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 \(M\) be a compact non-singular real affine algebraic variety and let \(A,B\) be open disjoint semialgebraic subsets of \(M\). Define \(Z= \overline {\overline A\cap \overline B}^Z\) (where \(\overline{\phantom{A\cap B}}^Z\) denotes the Zariski closure). The sets \(A,B\) are said generically separated if there exists a proper algebraic subset \(X\subset M\) and a polynomial function \(p\in {\mathcal P} (M)\) (or equivalently a regular function \(p\in {\mathcal R}(M))\) such that \(p(A-X)>0\) and \(p(B-X)<0\). The sets \(A,B\) are said separated if there exists \(p\in {\mathcal P}(M)\) such that \(p(A-Z)>0\) and \(p(B-Z)<0\). Very general results on the problem of polynomial separation for semialgebraic sets are known, for instance Bröcker solved the problem by reducing it to the separation of constructible sets in a space of orderings. Proofs involve powerful tools of real algebra and quadratic forms theory. We are interested in finding a finite number of geometric conditions equivalent to the separation of two open semialgebraic sets going towards an algorithmic solution of the problem. In this article we consider the case of a compact nonsingular algebraic variety \(M\) of dimension 3. Our main result is a criterion of separation working essentially when the Zariski boundaries of the sets \(A\) and \(B\) have only nonsingular normal crossings components. Up to desingularization, this criterion reduces the separation problem in dimension 3 to a separation problem on the Zariski boundaries of the sets, hence to a finite number of tests. It is a first step to prove the ``decidability'' of this problem. generically separated sets; threefolds; semialgebraic subsets; constructible sets; separation problem F. Acquistapace, F. Broglia, and E. Fortuna, A separation theorem in dimension 3, Nagoya Math. J. 143 (1996), 171 -- 193. Semialgebraic sets and related spaces, \(3\)-folds, Computational aspects of higher-dimensional varieties A separation theorem in dimension 3

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 1956, Abhyankar stated his local weak simultaneous resolution theorem for algebraic surfaces. By using this theorem, he was able to eliminate the use of Zariski's factorization theorem for proving the local uniformization theorem for algebraic surfaces (a step in Zariski's plan for proving resolution of singularities). The author thinks that generalizing the local weak simultaneous resolution theorem could be useful to show local uniformization for higher-dimensional varieties since Zariski's factorization theorem is not true for these varieties. In this sense, he gives a partial generalization which does not account for all possible rational ranks. The concrete result (section 3 of the paper) which is proved from a monomialization theorem for ordered semigroups (section 2) is the following: Let \(K\) be an \(n\)-dimensional function field over an algebraically closed field \(k\). Assume that either the characteristic of \(k\) is zero or that it is positive and \(n\leq 3\). Let \(L/K\) be a finite algebraic extension. Let \(v\) be a valuation of \(L/k\) with \(k\)-dimension zero, rational rank \(r\), and valuation ring \(V\). Let \((S,N)\subset L\) be an \(n\)-dimensional local ring over \(k\) which is birationally dominated by \(V\). Let \(R=S\cap K\), \(M=N\cap K\), \(U=V\cap K\) and \(Q=MS\). (1) If \(r=n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(Q\) is \(N\)-primary; (2) If \(r<n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(\text{ht} Q\geq r+1\). In particular, if \(r=n-1\), then for such a transform \(Q\) is \(N\)-primary. simultaneous resolution; valuations; monomialization; local uniformization; function field DOI: 10.1006/jabr.1996.7014 Valuations and their generalizations for commutative rings, Arithmetic theory of algebraic function fields, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Algebraic functions and function fields in algebraic geometry, Regular local rings Local weak simultaneous resolution for high rational ranks

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 theory of linear systems on graphs and tropical curves, introduced by Baker, has rapidly become an important area of mathematics bridging combinatorics and algebraic geometry. Some striking applications include a recent essentially optimal bound on the number of rational points on algebraic curves, by Katz and Zureick-Brown. The key property of these graphical linear systems (in addition to the fact that they are combinatorial and hence often more easily computed than their classical geometric counterparts) is the Specialization Inequality, which states that rank can only increase when tropicalizing. One thinks of this as a form of upper-semicontinuity, where the classical curve is viewed as the generic fiber of a family and the tropicalization somehow the special fiber. One also has upper-semicontinuity purely on the tropical side, where one has a natural topology on the moduli space of tropical curves (i.e., metric graphs). Fundamental to both these results is that the rank of a tropical linear system, as defined by Baker and Norine, is not the dimension of the obvious \(\mathbb{T}\)-module of sections, but rather is a purely combinatorial definition that they introduce (which coincides with one of the equivalent definitions on the classical side) Another step in this story is that rather than considering a single divisor/linear system at a time, one can consider an entire moduli space of them. Specifically, there is a graphical/tropical analogue of Brill-Noether loci, and these have been used by Payne and his collaborators to find new tropical proofs of the classical Brill-Noether and Gieseker-Petri theorems on the dimensions and smoothness of these loci. However, the dimensions of these loci do not vary upper-semicontinuously on the moduli space of tropical curves, as they do classically. There is an obvious obstruction given by separating edges, but even after contracting the corresponding loci in the moduli of tropical curves, the present papers finds counterexamples to upper-semicontinuity (and along the way, counterexamples to a conjecture of Luo on renk-determining sets as well). To rectify this, the authors introduce a new notion of dimension for the Brill-Noether loci. As with Baker-Norine rank, their notion is more of a direct combinatorial definition rather than the topological dimension of a \(\mathbb{T}\)-module. Remarkably, this fixes the previous issues as the authors prove a specialization and upper-semicontinuity result for their proffered definition. Surely the notion arrived at in the present paper is the correct one, and it will be interesting to see how it is used in future research, as the saga continues to unfold. tropical curves; metric graphs; linear systems; Brill-Noether; upper-semicontinuity Lim, C.M., Payne, S., Potashnik, N.: A note on Brill--Noether theory and rank determining sets for metric graphs. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/rnr233 , Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory) A note on Brill-Noether theory and rank-determining sets for metric graphs

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 normal, locally Noetherian scheme, \(G\) a Chevalley-Demazure group scheme with ``épinglage'' over \(S\). Fix a generic geometric point \(\overline\eta:\text{Spec} K\to S\), and let \(\rho_ K:G_ K \to Gl(V_ K) \subset \text{End}(V_ K)\) be a representation of \(G_ K\). An \(S\)- form of \(V_ K\) is a pair \(\bigl( V({\mathcal O}_ S),i \bigr)\), where \(V({\mathcal O}_ S)\) is a vector bundle over \(S\) equipped with a \(G\)-action and \(i:V({\mathcal O}_ S)_ K \to V_ K\) is an isomorphism respecting the action of the group scheme \(G\). The author classifies all \(S\)-forms of \(V_ K\) by constructing a mapping from the set of such forms to \(\text{Pic} S\) and then classifying the elements in the fibre of the map as locally free \({\mathcal O}_ S\)-modules equipped with a graded structure defined in terms of a Chevalley system of \(G_ K\). This characterization is extended to nonsplit group schemes \(G\) by using the rigidity of the épinglage structure to construct descent data. Here the fibres are classified by \(H^ 1\bigl(S_{\text{ét}},\Aut(\rho)\bigr)\). The author finishes by describing \(\Aut(\rho)\) as an extension of \(G_ m\) by \(H\), where \(H\) is a semidirect product of \(\text{ad}(G)\) by the discrete group of automorphisms of the épinglage structure. \(S\)-forms; automorphism group; Chevalley-Demazure group scheme; action of group scheme Group schemes, Representation theory for linear algebraic groups Representations of reductive group 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 Let \(X\) be a nondegenerate reduced closed subscheme in a projective space \(\mathbb{P}^N\) over an algebrically closed field \(k\). Let \(I\subseteq k[x_0,\cdots,x_N]\) be a homogeneous ideal generated by homogeneous polynomials of degree \(2\) such that \(\tilde{I}=\mathfrak{I}_X\). The authors call \(X\) to satisfy condition \textit{\(N_{2,p}\) scheme-theoretically} if \(I\) has linear syzygies at least up to the first \(p\)th steps. This definition generalizes the notation \(N_p\) for normal subschemes in [\textit{M. Green} and \textit{R. Lazarsfeld}, Compos. Math. 67, No. 3, 301--314 (1988; Zbl 0671.14010)]. One of the purposes of the paper is to see how the property \(N_{2,p}\) behaves under projection from a linear subvariety. Reducing to projection from a point, the authors consider the \textit{\(x_0\)-elimination ideal} of \(I\) in \(S=k[x_1,\cdots,x_N]\) which is \(J=I\cap S\). To study \(J\), the authors consider the \(S-\)module structure of \(I\) (which is an infinitely generated module) and apply the \textit{elimination mapping cone} construction as in [\textit{J. Ahn} and \textit{S. Kwak}, J. Algebra 331, No. 1, 243--262 (2011; Zbl 1232.14035)]. It is shown in Corollary 3.4 that the inner projection from a smooth point satisfies \(N_{2,p-1}\), provided \(X\) satisfies \(N_{2,p}\). Also Theorem 4.1 shows that the arithmetic depth is preserved under the inner projection from a smooth point. These results enable the authors to characterize varieties which satisfy condition \(N_{2,p}\) in some extremal cases in Theorem 4.3, namely if \(X\) is \(r\)-dimensional variety, \(N=r+e\), \(p=e\) or \(p=e-1\). Another interesting result of the paper is that the ideal \(I\), defined at the beginning, must have at least \(\text{codim}(X).p-\frac{p(p-1)}{2}\) generators, if \(X\) satisfies \(N_{2,p}\). The paper ends with some examples and open questions. linear syzygies; mapping cone theorem; inner projection; elimination ideal; arithmetic depth; Castelnuovo-Mumford regularity K. Han and S. Kwak, \textit{Analysis on some infinite modules, inner projection, and applications}, Trans. Amer. Math. Soc., 364 (2012), pp. 5791--5812. Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Varieties of low degree Analysis on some infinite modules, inner projection, and applications

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 field of positive chracteristic $p>0$ and $R$ a formal power series ring over $k.$ Let $A=R/I$ be a reduced Cohen-Macaulay ring of dimension $d\geq 1.$ Let also $\Omega/I$ be a canonical ideal of $A.$ Assume that $d\geq 2, \Omega\neq R$ and $J$ is a radical ideal defining the singular locus of $R/\Omega.$ The main result of the paper shows that if $J(\Omega ^{[p]}:\Omega)\nsubseteq \mathfrak{m}^{[p]},$ then $A$ is a strongly $F$-regular ring. Another result gives an explicit description of a generating morphism for the local cohomology module $H^2_{I_{n-1}(X)}(k[[X]])$, where $X$ is an $n\times (n-1)$ matrix of indeterminates over $k.$ As an application of the above results it is shown that, if $k$ is perfect and $p\geq 5,$ the generic determinantal ring $k[[X]]/I_{n-1}(X)$ is strongly $F$-regular. tight closure; strongly $F$-regular ring; local cohomology; determinantal ring Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Local cohomology and commutative rings, Linkage, complete intersections and determinantal ideals, Local cohomology and algebraic geometry, Determinantal varieties Strong \(F\)-regularity and generating morphisms of local cohomology 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 Theorem 0.2 (General Néron desingularization). Let \(u:A \to A'\) be a morphism of Noetherian rings. Then \(u\) is regular if and only if \(A'\) is a filtered inductive limit of smooth \(A\)-algebras. Theorem 0.2 gives an easy proof of the fact that the Artin approximation property holds for excellent Henselian local rings, but to show that the strong Artin approximation property holds for the same rings is essentially more complicated. It would be nice to have a kind of ``approximate'' desingularization which can give easy proofs also in this last case. This is provided by the following theorem which is in fact the purpose of our paper: Theorem 0.4. Let \((A,m)\) be a quasi-excellent local ring. Then \(A\) has the following property: (*) For every finite type \(A\)-algebra \(B\), there exists a function \(\nu: {\mathbb N}\to{\mathbb N}\) such that for every positive integer \(c\) and every \(A\)-algebra morphism \(v:B\to A/m^{\nu(c)}\), there exist a finite type smooth \(A\)-algebra \(C\) and two \(A\)-algebra morphisms \(t:B\to C\), \(w:C\to A/m^c\) such that (*1) \(C\) is smooth over \(B\) at \(\widetilde t^{-1}(D)\), \(D\) being the smooth locus of \(B\) over \(A\), (*2) the following diagram commutes: \[ \begin{matrix} B & @>v>> & A/m^{\nu(c)}\\ \downarrow & &\downarrow \\ C &@>w>> & A /m^c \end{matrix} \] where the left map is \(t\). excellent Henselian local rings; strong Artin approximation property; smooth locus; approximate desingularization 20. D. Popescu , Variations on Néron desingularization, in Sitzungsberichte der Berliner Mathematischen Gesselschaft (Berlin, 2001), pp. 143-151. Morphisms of commutative rings, Henselian rings, Singularities in algebraic geometry, Commutative Noetherian rings and modules Variations of Néron 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 The authors study some actions on the set \(M_{n,m}\) of matrices with coefficients in the ring of formal power series \(R:=K[[x_1,\ldots, x_n]]\), where \(K\) is a field. They study the action of \(G:=Gl_n(R)\) on \(M_{n,m}\) by multiplication from the left (or from the right, or from both sides). Then they give a condition to insure the finitely determinacy of a matrix \(A\). One says that a matrix \(A:=(a_{ij}(x))\) is finitely determined (resp. \(G\)-finitely determined) if, for every matrix \((b_{ij}(x))\) whose entries coincide with those of \(A\) up to some high power of the maximal ideal of \(R\), there exists an automorphism \(\varphi\) of \(R\) such that \((\phi(b_{ij}(x)))=(a_{ij}(x))\) (resp. such that \((\phi(b_{ij}(x)))\) belongs to the orbit of \(A\) under the action of \(G\)). The condition given in this paper is expressed in terms of the tangent image of the orbit map. In characteristic zero, they prove that this condition is not only sufficient but also necessery. These results generalize previous classical results concerning the finite determinacy of power series or of vectors of power series. matrices with power series coefficients; finite determinacy Singularities in algebraic geometry, Formal power series rings, Matrices over function rings in one or more variables On finite determinacy for matrices of power series

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 give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Theorem 1.1 determines the class in \(\Gamma_D\) of the local cohomology groups of each \(D_p\), thus generalizing the main result of [\textit{C. Raicu} and \textit{J. Weyman}, Algebra Number Theory 8, No. 5, 1231--1257 (2014; Zbl 1303.13018)] which addresses the case \(p = n\). For nonsquare matrices, Theorem 1.1, together with the fact that \(\bmod_{\mathrm{GL}}(D_X )\) is semisimple, gives a description of Lyubeznik numbers. Specializing our results to a single iteration, they determine the Lyubeznik numbers for all generic determinantal rings and prove the vanishing of a range of local cohomology groups. Next, they use the quiver description of the category \(\bmod_{\mathrm{GL}}(D_X )\) in conjunction with the vanishing results to provide an inductive proof of Theorem 1.6. determinantal varieties; local cohomology; Lyubeznik numbers; equivariant \(\mathcal{D}\)-modules Local cohomology and commutative rings, Homological functors on modules of commutative rings (Tor, Ext, etc.), Determinantal varieties Iterated local cohomology groups and Lyubeznik numbers for determinantal 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 \(\{f=0,0\}\) be a germ of a hypersurface singularity in \(({\mathbb{C}}^{n+1},0)\) with a 1-dimensional singular set \(\Sigma\). If x is a generic linear form, a little deformation \(f+\epsilon x^ N\) (\(\epsilon\) little, N big) has an isolated singularity. Let S be a local irreducibe component of \(\Sigma\) at 0. Along S-\(\{\) \(0\}\), f can be viewed as a \(\mu\)-constant deformation of the transversal section (which has an isolated singularity). These singularities have a monochromy (called horizontal) and the local system over S-\(\{\) \(0\}\) defines another monodromy (called vertical). The main theorem of this paper relates the characteristic polynomials of the monodromies of f, \(f+\epsilon x^ N\), the vertical and the horizontal monodromies. The methods, polar curves and carroussel, are essentially topological. Almost simultaneously, M. Saito has proved in this situation the Steenbrink conjecture [\textit{M. Saito}, Math. Ann. 289, 703-716 (1991)], which relates the spectra of f and of \(f+\epsilon x^ N\). The monodromy is characterized by the values mod \({\mathbb{Z}}\) of the spectrum. This better result is proved by the Saito theory of mixed Hodge modules. germ of a hypersurface singularity; characteristic polynomials; Steenbrink conjecture; monodromy D. Siersma, ''The Monodromy of a Series of Hypersurface Singularities,'' Comment. Math. Helv. 65, 181--197 (1990). Complex surface and hypersurface singularities, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Deformations of complex singularities; vanishing cycles, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The monodromy of a series of hypersurface 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 long paper, the author studies local systems on smooth (not necessarily complete) curves \(X\) by working in the framework of analytic étale topology of \textit{V. Berkovich}. The base field \(k\) is a field of characteristic \(0\) complete with respect to a non-archimedean metric and with a residue field of positive characteristic. The notion of an analytic local fundamental group at a point of \(X\) is defined so that a locally constant sheaf on a small punctured disc at the point gives a continuous representation of this group. A canonical quotient of this group is constructed which is the analogue of the local differential Galois group classifying connections with poles of finite orders. An analytic Swan conductor is defined in terms of a canonical higher ramification filtration on the local fundamental group. It is proved that the Swan conductor of a finite rank representation is an integer (an analogue of the Hasse-Arf theorem). The author defines and studies meromorphically ramified local systems on an open curve with certain bounds on the ramification at the points at infinity. The main tool is the Fourier transform. It is shown that the cohomology of a meromorphically ramified local system has finite rank. A formula of Grothendieck-Ogg-Shafarevich type is proved for all meromrphically ramified sheaves on any smooth open curve using a local Morse-theoretic argument. Finally the author defines a Galois module \(\Gamma (q)\) of a quadratic form \(q\) which descends to a homomorphism from the Witt group of \(k\) to the group of isomorphism classes of rank \(1\) \( \ell-\)adic Galois modules. This is inspired by classical work of A. Weil. étale analytic topology; meromorphic local systems; local differential Galois group; analytic Swan conductor; canonical higher ramification filterion; Fourier transform; Witt group Lorenzo Ramero, On a class of étale analytic sheaves, J. Algebraic Geom. 7 (1998), no. 3, 405 -- 504. Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On a class of étale analytic sheaves

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 number field and \(G= GL_ n(K)\) the group of invertible \(n\times n\)-matrices. Each matrix in \(G\) defines a linear transformation on \(V:= K^ n\) by acting on the right to row vectors. Let \({\mathbf v}_ 1= (1,0,\dots,0), \dots,{\mathbf v}_ n\) denote the standard basis of \(V\) and let \(V_ i\) be the vector space with basis \({\mathbf v}_ 1,\dots, {\mathbf v}_ i\). For given integers \(d_ 1,\dots, d_ l\) with \(0< d_ 1<\dots <d_ l<n\), let \(P\) be the subgroup of \(G\) consisting of those linear transformations mapping the subspaces \(V_{d_ 1},\dots, V_{d_ l}\) to itself. Then \(P\setminus G\) parametrises the nested sequences of linear subspaces \(S_ 1\subset S_ 2\subset \cdots\subset S_ l\) of \(V\) with \(\dim S_ i= d_ i\) for \(i=1,\dots, l\). \(P\setminus G\) may be considered as the set of \(K\)- rational points of a projective variety \({\mathcal V}\), called a flag variety. The author defines a (non-logarithmic) height \(H\) on \({\mathcal V}(K)\) by using a suitably metrised ample line bundle (in fact the anti- canonical bundle of \({\mathcal V})\). Let \(N({\mathcal V}(K),B)\) denote the number of points of \({\mathbf x}\in {\mathcal V}(K)\) with \(H(x)\leq B\). \textit{J. Franke}, \textit{Yu. I. Manin}, and \textit{Yu. Tschinkel} [Invent. Math. 95, 421-435 (1989; Zbl 0674.14012)] derived an asymptotic formula of the type \(N({\mathcal V}(K), B)\sim cB\log^{l-1}(B)\) as \(B\to\infty\), using results on the analytic continuation of certain \(L\)-series. The author derives a similar asymptotic formula in a much less involved way, using geometry of numbers and lattice point counting arguments. The author's method of proof resembles that of \textit{S. H. Schanuel} who derived an asymptotic formula for \(N({\mathcal V}(K), B)\) for \({\mathcal V}= \mathbb{P}^ m\) [Bull. Soc. Math. Fr. 107, 433-449 (1979; Zbl 0428.12009)]. rational points; projective variety; flag variety; height; asymptotic formula J.\ L. Thunder, Asymptotic estimates for rational points of bounded height on flag varieties, Compos. Math. 88 (1993), 155-186. Arithmetic algebraic geometry (Diophantine geometry), Geometry of numbers, Grassmannians, Schubert varieties, flag manifolds, Rational points, Arithmetic varieties and schemes; Arakelov theory; heights Asymptotic estimates for rational points of bounded height on flag 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 Let \(S\) be a complete non singular algebraic surface and \(D\in\text{Div}(S)\) with normal crossing. Then the pair \((S,D)\) determines a weighted graph which carries some information about the surface \(S\backslash D\). If \(D\) does not have normal crossings then one has to blow-up \(S\) at the ``bad'' points of \(D\). In this paper it is developed a graph theory which relates the desingularization process to graph-theoretic devices called ``local trees''. Application of the methods to the classification of birational morphisms of the affine plane with one or two fundamental points is given. plane curves; divisor on algebraic surface; local trees; weighted graph; desingulariztion Daigle, D.: Local trees in the theory of affine plane curves, J. math. Kyoto univ. 31, 593-694 (1991) Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, Enumerative problems (combinatorial problems) in algebraic geometry Local trees in the theory of affine 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 Let \(R\) be a complete discrete valuation ring of equi-characteristic zero with fraction field \(K\). Let \(X\) be a connected smooth projective variety of dimension \(d\) over \(K\), and let \(L\) be an ample line bundle over \(X\). We assume that there exist a regular strictly semistable model \(\mathscr{X}\) of \(X\) over \(R\) and a relatively ample line bundle \(\mathscr{L}\) over \(\mathscr{X}\) with \(\mathscr{L}|_{X}\cong L\). Let \(S(\mathscr{X})\) be the skeleton associated to \(\mathscr{X}\) in the Berkovich analytification \(X^{\mathrm{an}}\) of \(X\). In this article, we study when \(S(\mathscr{X})\) is faithfully tropicalized into tropical projective space by the adjoint linear system \(|L^{\otimes m}\otimes\omega_X|\). Roughly speaking, our results show that if \(m\) is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of \(S(\mathscr{X})\). Geometric aspects of tropical varieties, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Embeddings in algebraic geometry Effective faithful tropicalizations associated to adjoint linear 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 \(X_{s,(0)}\) be the blow-up of \(\mathbb P^n\) at s points in general position. In the previous manuscript [``Positivity of divisors on blown-up of projective spaces. I'', Preprint, \url{arXiv:1506.04726}], the same authors studied Fujita's conjecture about global generation (and very ampleness) and prove it for \(s \le 2n\). In the paper under review, the authors investigate further positivity properties of divisors on \(X_{s,(0)}\) and other spaces \(X_{s,(r)}\) obtained as subsequent blow-ups along linear cycles spanned by some of the \(s\) points. More precisely, let \(D\) be a general element in the linear system \[\vert dH - \sum_{i=1}^s m_iE_i \vert\] on \(X_{s,(0)}\), where \(H\) is a hyperplane section, \(E_i\) are the exceptional divisors and \(d, m_1,\ldots, m_s\) are positive integers. Let \(D_{(r)}\) be the strict transform of \(D\) on \(X_{s,(r)}\). The main result of the paper (Theorem 2.1) is a description of when \(D_{(r)}\) is semi-ample or globally generated. The proof is based on the study of a resolution of singularities of the base locus of \(D\) and on vanishing theorems . As an application the authors show that the abundance conjecture holds true for infinitely many families of explicitly constructed log canonical pairs \((X_{s,(0)}, D)\). It is known that the space \(X_{n+2, (n-2)}\) obtained as the iterated blow-up along all linear subspaces of \(\mathbb P^n\) spanned by the \(s\) points is isomorphic to the moduli space \(\overline{ \mathcal M}_{0,n+3}\) of rational stable pointed curves. As another application of their results, the authors show that F-conjecture holds true for all divisors on \(X_{n+2, (n-2)}\) that are strict transforms of an effective divisor on \(X_{n+2,(0)}\). blow-up of projective space; globally generated divisors; abundance conjecture; F-conjecture Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Hypersurfaces and algebraic geometry Positivity of divisors on blown-up projective spaces. II.

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 characterize ideals \(I\) in the power series ring \(R = K [[x_1, \ldots, x_s]]\) that are finitely determined up to contact equivalence by proving that this is the case if and only if \(I\) is an isolated complete intersection singularity, provided \(\dim(R / I) > 0\) and \(K\) is an infinite field (of arbitrary characteristic). Here two ideals \(I\) and \(J\) are contact equivalent if the local \(K\)-algebras \(R / I\) and \(R / J\) are isomorphic. If \(I\) is minimally generated by \(a_1, \ldots, a_m\), we call \textit{I finitely contact determined} if it is contact equivalent to any ideal \(J\) that can be generated by \(b_1, \ldots, b_m\) with \(a_i - b_i \in \langle x_1, \ldots, x_s \rangle^k\) for some integer \(k\). We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left-right equivalence on the ring \(M_{m, n}\) of \(m \times n\) matrices \(A\) with entries in \(R\) and we show that the Fitting ideals of a finitely determined matrix in \(M_{m, n}\) have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in \(R\). Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture. singularities; finite determinacy; positive characteristic; algebraic group action; inseparable orbit action; specialization of power series; complete intersections Formal power series rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Local complex singularities Finite determinacy of matrices and 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 A henselian two-dimensional local field \(\Lambda\) is, by definition, an excellent henselian discrete valuation field whose residue field is a henselian discrete valuation field with finite residue field. If \(A\) is a two-dimensional excellent normal henselian local ring with finite residue field, then, for any prime \(\mathfrak p\) of height one in \(A\), the field of fractions \(K_{\mathfrak p}\) of the henselization of \(A\) at \(\mathfrak p\) is a two-dimensional local field. And, conversely, every two-dimensional local field arises in this way. The Brauer group of a two-dimensional local field has been extensively studied by \textit{K. Kato} in his three papers ``A generalization of local class field theory by using \(K\)-groups'' [(I) J. Fac. Sci., Univ. Tokyo, Sec. I A 26, 303--376 (1979; Zbl 0428.12013); (II) ibid. 27, 602--683 (1980; Zbl 0463.12006); (III) ibid. 29, 31--43 (1982; Zbl 0503.12004)]. Using the results of Kato, the author describes the Brauer group \(\mathrm{Br}(K)\) of the field of fractions \(K\) of a henselian ring \(A\) as above. Among several interesting results, he constructs a canonical pairing \(\mathrm{Br}(X)\times\mathrm{Pic}(X)\to\mathbb Q/\mathbb Z\) where \(X\) is the punctured spectrum of \(A\) and shows that it is in fact a perfect pairing of finite abelian groups. In an appendix he defines a similar perfect pairing when \(X\) is a projective smooth geometrically connected curve over a complete discrete valuation field with finite residue field. This, in the case of a field \(k\) of positive characteristic, improves upon a result of \textit{S. Lichtenbaum} in [Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)]. Picard group; duality; Henselian two-dimensional local field; Brauer group Shuji Saito, ''Arithmetic on two-dimensional local rings,''Invent. Math.,85, 379--414 (1986). Brauer groups of schemes, Arithmetic ground fields for curves, Henselian rings, Valued fields Arithmetic on two dimensional 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 The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes. An optimal Hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of R. Fröberg and A. Iarrobino. We discuss these conjectures from a geometric point of view. They give, for each such collection, a function (based on dimension, number of points, and order of each neighbourhood) which should serve as an upper bound to its Hilbert function (``weak conjecture''). The ``strong conjecture'' predicts when the upper bound is sharp, in the case of equal order throughout. In general we refer to the equality of the Hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the ``strong hypothesis''. We interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points. We develop techniques and insight toward the conjectures' verification and refinement. The main result gives an upper bound on the Hilbert function of a collection of infinitesimal neighbourhoods in \(\mathbb P^n\) based on Hilbert functions of certain such subschemes of \(\mathbb P^{n-1}\). Further, equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand. It follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis. This observation is then applied to show that the weak conjecture does hold valid in \(\mathbb P^n\) for \(n \leqslant 3\). The main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in \(\mathbb P^2\) and, further, \(\mathbb P^2\) presents special exceptional cases. Consequences of the main result in higher dimension are then examined. We note, then, that the full weight of the strong conjecture (and validity of the strong hypothesis) are not necessary toward using the main theorem in the next dimension. We end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information: namely, on the structure of the minimal free resolution of an ideal generated by linear forms. Chandler, K, The geometric interpretation of fröberg-iarrobino conjectures on infinitesimal neighbourhoods of points in projective space, J. Algebra, 286, 421-455, (2005) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The geometric interpretation of Fröberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in 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 We consider the limiting behavior of \textit{discriminants}, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety \(X\) and linear systems on \(X\). These are connected -- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological. Grothendieck ring; stabilization; discriminant; configuration spaces; hypersurfaces; motivic zeta functions Arcs and motivic integration, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Fibrations, degenerations in algebraic geometry Discriminants in the Grothendieck 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 Let \textit{R} be a regular local ring. Let \textbf{G} be a reductive group scheme over \textit{R}. A well-known conjecture due to Grothendieck and Serre assertes that a principal \textbf{G}-bundle over \textit{R} is trivial, if it is trivial over the fraction field of \textit{R}. In other words, if \textit{K} is the fraction field of \textit{R}, then the map of non-abelian cohomology pointed sets \[ H_{ét}^1(R,\mathbf{G})\rightarrow H_{ét}^1(K,\mathbf{G}), \] induced by the inclusion of \textit{R} into \textit{K}, has a trivial kernel. \textit{The conjecture is solved in positive for all regular local rings contaning a field}. More precisely, if the ring \textit{R} contains an infinite field, then this conjecture is proved in a joint paper due to \textit{R. Fedorov} and \textit{I. Panin} [Publ. Math., Inst. Hautes Étud. Sci. 122, 169--193 (2015; Zbl 1330.14077)]. If the ring R contains a finite field, then this conjecture is proved in 2015 in a preprint due to \textit{I. Panin} [``Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a finite field'', Preprint, \url{https://www.math.uni-bielefeld.de/LAG/man/559.pdf}] which can be found on preprint server Linear Algebraic Groups and Related Structures. A more structured exposition can be found in \textit{I. Panin}'s preprint of the year 2017 [``Proof of Grothendieck-Serre conjecture on principal bundles over regular local rings containing a finite field'', Preprint, \url{arXiv:1707.01767}]. This and other results concerning the conjecture are discussed in the present paper. We illustrate the exposition by many interesting examples. We begin with couple results for complex algebraic varieties and develop the exposition step by step to its full generality. linear algebraic groups; principal bundles; affine algebraic varieties Linear algebraic groups over arbitrary fields, Linear algebraic groups over adèles and other rings and schemes, Cohomology theory for linear algebraic groups, Group schemes On Grothendieck-Serre conjecture concerning principal 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 The proof of the following result is given. Let V be a compact nonsingular real algebraic set. Then there exists an algebraic resolution \(\pi\) : \(\tilde V\to V\) such that \(H_ k(\tilde V;{\mathbb{Z}}_ 2)\) is generated by components of k-dimensional compact nonsingular algebraic subsets (precisely, stable algebraic subsets of \(\tilde V\) for all k. - The reason for this result not to be a direct consequence of Hironaka's resolution theorem is that not all homology cycles of V may be represented by algebraic subsets. real algebraic set; algebraic resolution; stable algebraic subsets Akbulut S., King H.: A resolution theorem for homology cycles of real algebraic varieties. Invent. Math. 79, 589--601 (1985) Real algebraic and real-analytic geometry, Realizing cycles by submanifolds, Global theory and resolution of singularities (algebro-geometric aspects) A resolution theorem for homology cycles of 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 Let A be a (Noetherian) local ring of dimension \(d\geq 1\), having maximal ideal \({\mathfrak m}\). We say that Hochster's monomial conjecture [\textit{M. Hochster}, Nagoya Math. J. 51, 25-43 (1973; Zbl 0268.13004), conjecture 1] holds for the system of parameters (s.o.p.) \(x_ 1,...,x_ d\) for A if, for every integer \(n\geq 0\), we have \(x_ 1^ n...x_ d^ n\not\in Ax_ 1^{n+1}+...+Ax_ d^{n+1}.\) In Mathematika 29, 296-306 (1982; Zbl 0523.13001), the authors showed that the monomial conjecture can be formulated in terms of their generalized fractions introduced by them in Mathematika 29, 32-41 (1982; Zbl 0497.13006). - The paper under review uses this generalized fraction approach to the monomial conjecture and the fact that the local cohomology module \(H_{{\mathfrak m}}^{d-1}(A)\) is Artinian to prove among other things, the following result. Suppose that (d\(\geq 2\) and) the monomial conjecture is known to hold for all systems of parameters in all local rings of dimension d-1. Then there exists a positive integer s, depending only on \(H_{{\mathfrak m}}^{d-1}(A)\), such that, for each s.o.p. \(x_ 1,...,x_ d\) for A, the monomial conjecture holds for \(x_ 1^ s,x_ 2,...,x_ d\). Noetherian local ring; monomial conjecture; system of parameters; s.o.p.; generalized fractions; local cohomology module DOI: 10.1016/0021-8693(85)90128-0 Local rings and semilocal rings, Rings of fractions and localization for commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Commutative Noetherian rings and modules, Complexes, Local cohomology and algebraic geometry Generalized fractions and the monomial 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 \((R,m)\) be a complete normal local domain of dimension 2. The set of complete m-primary ideals of R, m(R), has a natural semigroup structure. Assume that \(k=R/m\) is algebraically closed. Then we have the following results: (1) m(R) has unique factorization \(\Leftrightarrow\) R is a UFD; (2) R satisfies condition (N) \(\Leftrightarrow\) the class group Cl(R) is torsion. The implication \(\Leftarrow\) of (1) was proved by \textit{J. Lipman} in Publ. Math., Inst. Hautes Étud. Sci. 36 (1969), 195-279 (1970; Zbl 0181.489). The implication \(\Leftarrow\) of (2) was proved by \textit{H. Göhner} [J. Algebra 34, 403-429 (1975; Zbl 0308.13023)]. Condition (N) was introduced by \textit{H. T. Muhly} and \textit{M. Sakuma} [J. Lond. Math. Soc. 38, 341-350; 494 (1963; Zbl 0142.288)]. The other two implications are proved in the present paper. (These results are extended to the case where k is not algebraically closed in the part II of the paper under review [Invent. Math. 98, No.1, 59-74 (1989)]. When k is algebraically closed of characteristic zero, it is proved that R has a rational singularity if and only if R satisfies condition (N). An example is given of a nonrational singularity R defined over an algebraically closd field of characteristic \(p>0\) such that R satisfies condition (N). Finally several cases in dimension greater than 2 are considered. semifactoriality; complete ideals; unique factorization; UFD; condition (N); class group; rational singularity S.D. Cutkosky, On unique and almost unique factorization of complete ideals , Amer. J. Math. 111 (1989), 417-433. JSTOR: Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Singularities of surfaces or higher-dimensional varieties On unique and almost unique factorization of complete 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 Let \((R,m)\) be a regular local ring of dimension \(n\) and let \(A=R/I\) be a quotient of \(A\). Let \(\lambda_{a,i}=e(H_m^a(H_I^{n-i}(R)))\) be the invariant introduced and studied by Lyubeznik. The main result of the paper provides the descrption of this invariant in the case that \(A\) has reasonable singularities. local cohomology; characteristic \(p\); perverse sheaves Blickle, M.; Bondu, R., Local cohomology multiplicities in terms of étale cohomology, Ann. Inst. Fourier, 55, 2239-2256, (2005) Local cohomology and algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Local cohomology multiplicities in terms of étale cohomology

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 Zariski has shown that the equisingularity class of a plane algebroid branch (i.e. the germ of an irreducible algebroid curve) with local ring (A,\({\mathfrak m})\) is determined by the Puiseux-coefficients or - equivalently - by the semigroup \(\Gamma\) of the natural valuation of the normalization \(\bar A\) of A. The author gives a third characterization of this equisingularity class using the ``group of relative units'' of A, i.e. the quotient of the group of the units of \(\bar A={\mathbb{C}}[[t]]\) modulo the group of units of A, which has an algebraic structure (see theorem I.1.4) defined by certain polynomials. In the case of plane branches the degrees of these polynomials characterize the equisingularity class of the branch associated to A (see I.2.13 and also I.2.12 for the more general case). Moreover the relations between the A- algebra \(End_ A({\mathfrak m}^ p)\) and the semigroup associated to the branch are considered. equisingularity; plane algebroid branch; Puiseux-coefficients; valuation of the normalization; group of relative units Families, moduli of curves (algebraic), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings Sur la classification des branches. (On classification of branches)

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 present a resolution strategy that uses a modification of Villamayor's algorithm [\textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér. (4) 22, 1--32 (1989; Zbl 0675.14003)] as a subroutine and combines resolutions of irreducible (or at least equidimensional) components of a given algebraic set \(X\subset W\) to compute an embedded resolution of singularities of \(X\). The arising algorithm extends the scope of Villamayor's algorithm from equidimensional algebraic sets to the general case. The ideas also serve well in improving the efficiency of resolutions, using the prime ideal decomposition of the (radical) vanishing ideal of \(X\). Resolution of singularities Symbolic computation and algebraic computation, Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of algebraic surfaces Efficient desingularization of reducible algebraic sets

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 coadjoint orbits of affine algebraic linear supergroups \(G\). They adopt the point of view of (representable) functors of points for these groups, the domain category being here the category \((\text{salg}_k)\) of superalgebras over a commutative ring \(k\). Extending results from the classical (i.e. non-super) case, they show that the Lie algebra \({\mathfrak g}\) of a linear algebraic supergroup \(G\) gives rise to a representable functor (here \(k\) is a field). As in the classical case, the functor of points of the coadjoint orbit through a point \(X_0\in{\mathfrak g}^*\) \[ {\mathcal C}_{X_0}: (\text{salg}_k)\to (\text{sets }), A\mapsto\{\text{Ad}_g^*X_0\mid\forall g\in G(A)\} \] is not representable in general. The way-out is a sheafification of \({\mathcal C}_{X_0}\), which is explicitly constructed (for \(k={\mathbb C}\)) in terms of supersymmetric extensions \(\hat{p}_i\) of the homogeneous Chevalley polynomials \(p_i\), \(i=1,\ldots,l\) (where \(l\) is the rank of the group \(G({\mathbb C})\)) - recall that the orbit in \({\mathfrak g}({\mathbb C})^*\) of a regular semisimple element \(X_0\) is characterized by the \(p_i\). Here the authors restrict their study to orbits in a simple Lie superalgebra of type \({\mathfrak s}{\mathfrak l}_{m| n}\) or \({\mathfrak o}{\mathfrak s}{\mathfrak p}_{m| n}\) in order to benefit from a non-degenerate Cartan-Killing form and invariant polynomials \(p_i\). The sheafification is then represented by \(A_{X_0}={\mathbb C}[{\mathfrak g}]/{\mathcal I}\) where \({\mathcal I}=(\hat{p}_1-c_1,\ldots,\hat{p}_l-c_l)\) with \(c_i=p_i(X_0)\). In the last part, the authors consider deformation quantization of the constructed orbits. The main result is that \(U_h\,/\,{\mathcal I}_h\) is a formal deformation quantization of \({\mathbb C}[{\mathfrak g}^*]\,/\,{\mathcal I}\) where \(U_h\) is the universal enveloping algebra of \({\mathfrak g}_h:=({\mathfrak g},h[,])\), and \({\mathcal I}_h\) has the same form as \({\mathcal I}\), but featuring a supersymmetrized version of \(\hat{p}_i\) - as in the classical case, the supersymmetrization \(\tau\) identifies \({\mathbb C}[{\mathfrak g}^*]\) and \(U({\mathfrak g})\). affine algebraic supergroups; formal deformation quantization; coadjoint orbits Fioresi, R.; Lledó, M. A., On algebraic supergroups, coadjoint orbits, and their deformations, Comm. Math. Phys., 245, 1, 177-200, (2004) Quantum groups (quantized enveloping algebras) and related deformations, Affine algebraic groups, hyperalgebra constructions, Supervarieties, ``Super'' (or ``skew'') structure, Deformation quantization, star products, Supermanifolds and graded manifolds On algebraic supergroups, coadjoint orbits and their deformations

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 prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point \(\alpha\) where the small orbit by a polynomial \(f\) is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in \(\mathcal{S}_\alpha\) when everything is defined over the algebraic numbers and the degree \(d\) of \(f\) is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of \(\alpha\) if \(|f^{\circ n}(\alpha )|_v \rightarrow \infty\) at a finite place \(v\) of good reduction co-prime to \(d\). polynomial dynamical systems; Mordell-Lang conjecture; small orbits Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Local ground fields in algebraic geometry Polynomial dynamics and local analysis of small and grand 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 Let \(\mathbb{K}\) be an infinite field and \(P\) either \(\mathbb{K}[x_1,\dots,x_n]\) or \(\mathbb{K}[[x_1,\dots,x_n]]\) (formal power series). Let \(\hat{P} = \Hom_{P_0}(P,E)\), where \(E\) is the injective hull of \(\mathbb{K}\) over \(P_0\); we get a correspondence between graded ideals \(I\) of \(P\) and \(P\)-submodules \(I^\perp\) of \(\hat{P}\), under which the ideals \(I\) for which \(P/I\) is Artinian correspond to finitely generated submodules \(I^\perp\); \(I^\perp\) is known as the inverse system of \(I\) (by Macaulay). This correspondence as been extended to the case of positive dimension and such generalization is studied here, by using the varius socles in the inverse system. The main result is an explicit description of inverse limits of Macaulay's inverse systems, obtained by dividing out powers of a linear regular sequence. Macaulay inverse system; Matlis duality; Rees isomorphism Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Power series rings, Secant varieties, tensor rank, varieties of sums of powers Inverse limits of Macaulay's inverse 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 \(u:\quad A\to A'\) be a morphism of noetherian rings. Then \((i)\quad u\quad is\) regular iff it is a filtered inductive limit of finite type smooth morphisms. Under some separability conditions this result was the subject of our previous paper [Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008)] on which we rely now. Let \({\mathfrak a}\subset A\) be a proper ideal and \(\hat A\) the completion of A in the \({\mathfrak a}\)-adic topology. Using (i) it follows: \((ii)\quad if\) (A,\({\mathfrak a})\) is henselian and \(A\to \hat A\) is regular then every finite system of polynomials over A has its set of solutions in A dense with respect to the \({\mathfrak a}\)-adic topology in the set of its solutions in \(\hat A.\) - When A is local and \({\mathfrak a}\) is maximal then this result is a positive answer to one of M. Artin's conjectures [\textit{M. Artin}, Actes Congr. internat. Math. 1970, Vol. 1, 419-423 (1971; Zbl 0232.14003)]. As a consequence of (ii) it follows: \((iii)\quad the\) completion of an excellent henselian factorial local ring is factorial, too. - Also from (ii) we get a complete solution of so called approximation on nested subrings, i.e.: \((iv)\quad Let\) k be a field, \(k<X>\) the algebraic power series ring in \(X=(X_ 1,...,X_ r)\) over k, f a finite system of polynomials over \(k<X>\) and \(\hat y=(\hat y_ 1,...,\hat y_ n)\in k[[X]]^ n\) a formal solution such that \(\hat y_ i\in k[[X_ 1,...,X_{s_ i}]]\), \(1\leq i\leq n\) for some positive integers \(s_ i\leq r\). Then there exists a solution \(y=(y_ 1,...,y_ n)\) of f in \(k<X>\) such that \(y_ i\in k<X_ 1,...,X_{s_ i}>\), \(1\leq i\leq n\). regular morphism of noetherian rings; completion of an excellent henselian factorial local ring; approximation on nested subrings Popescu, D., General Néron desingularization and approximation, Nagoya Math. J., 104, 85-115, (1986) Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), Commutative Noetherian rings and modules, Complete rings, completion, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Henselian rings General Néron desingularization and approximation

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 \((A,m)\) be a regular local ring and \(M,N\) finitely generated \(A\)-modules, \[ \text{length}(M\otimes N)<\infty. \] Consider the intersection multiplicity \[ \chi(M,N): =\sum(-1)^i \text{length (Tor}^A_i(M,N)). \] Serre's conjecture on intersection multiplicities says that \(\chi(M,N)\geq 0\) with equality iff \[ \dim M+\dim N<\dim A. \] One can easily reduce the proof to the case when \(A\) is complete. Serre proved the conjecture when \(A\) is a power series ring over a field or a complete discrete valuation ring (d.v.r. for short) using ``complete Tor'' and the method of diagonalization. In the ramified case, the positivity part of the conjecture is still open. In this paper the author gives a new proof of Serre's theorem. His approach is quite different. He assumes firstly that \((A,m)\) is essentially of finite type over a field \(k\) or an excellent d.v.r. \((V,\pi)\) (and that \(A/m\) is separably generated over \(V/\pi V)\). Using a structure theorem for smoothness for such regular local rings due to \textit{B. S. Nashier} [J. Algebra 85, 287-302 (1983; Zbl 0527.13008)], the author reduces quickly this case to the case when \(A=k[X_1, \dots, X_d]_{(X_1,\dots, X_d)}\) or \(A=V[X_1, \dots, X_d]_{(\pi,X_1, \dots, X_d)}\). In this latter case one can use the method of diagonalization. The complete case is reduced to the above ``geometric'' case via M. Artin's approximation theorem. Nashier's result, which is the key point of this approach, is an extension of a result of \textit{H. Lindel} [Invent. Math. 65, 319-323 (1981; Zbl 0477.13006)] and was generalized by the author [\textit{S. P. Dutta}, J. Algebra 171, No. 2, 370-382 (1995; Zbl 0817.13013)]. Similar proofs are sketched for D. Quillen's theorem on Chow groups and its extension by \textit{H. Gillet} and \textit{M. Levine} [J. Pure Appl. Algebra 46, 59-71 (1987; Zbl 0685.14008)]. smoothness for regular local rings; Serre's conjecture on intersection multiplicities; Artin's approximation theorem; Chow groups Dutta, S. P., \textit{A theorem on smoothness-bass-Quillen, Chow groups and intersection multiplicity of Serre}, Trans. Amer. Math. Soc., 352, 1635-1645, (2000) Homological conjectures (intersection theorems) in commutative ring theory, Regular local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Étale and flat extensions; Henselization; Artin approximation, (Equivariant) Chow groups and rings; motives A theorem on smoothness. Bass-Quillen, Chow groups and intersection multiplicity of Serre

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 resolution of singularities in characteristic 0 was solved by Hironata at the beginning of the sixties of the last century. Much later algorithmic methods have been developed for solving this problem. The study of algorithmic equiresolution started about 15 years ago. Some basic results in this direction can be found in the article of \textit{S. Encinas, A. Nobile} and \textit{O. E. Villamayor} [Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 607--648 (2003; Zbl 1076.14020)]. Here families of ideals or of embedded schemes, parametrized by smooth varieties are studied. The equiresolution proposed required that the centers for the transformations are smooth over the parameter variety (condition AE) or required the local constancy of a certain invariant associated to each fibre (condition \(\tau\)). In this paper a definition of equiresolution is proposed for families parametrized by not necessarily reduced schemes, called condition E. Other approaches are proposed, condition A, C and F. Condition A corresponds to AE mentioned above. The main objective of the paper is to prove that when the parameter space is regular all these conditions are quivalent. Assuming the properness of certain projections it is proved that they are also equivalent to \(\tau\) mentioned above. resolution algorithm; embedded variety; coherent ideal; basic object Nobile, A.: Simultaneous algorithmic resolution of singularities, Geom. dedic. 163, 61-103 (2013) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, fibrations in algebraic geometry Simultaneous algorithmic 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 The problem of existence and construction of a resolution of singularities is one of the central tasks in algebraic geometry: Given a variety \(X\) over a field \(K\), a resolution of singularities of \(X\) is a proper birational morphism \(\pi:Y\to X\) such that \(Y\) is a non--singular variety. If \(K\) is a field of characteristic \(0\) the existence of resolution of singularities has been proved by \textit{H. Hironaka} [Ann. Math. (2) 79, 109--326 (1964; Zbl 0122.38603)]. His proof is highly non--constructive. Important contributions to find algorithms for the resolution of singularities have been made independently by the groups of \textit{E. Bierstone} and \textit{P. Milman} [Invent. Math. 128, 207--302 (1997; Zbl 0896.14006)] and of O. Villamayor and S. Encinas. The paper contains a short and simplified proof of the desingularization theorem (slightly stronger than Hironaka's version: strong factori\-zing desingularization). The resolution process is given by a sequence of blowing ups at carefully chosen centers. The choice of these centers is achieved without use of the Hilbert--Samuel function and Hironaka's notion of normal flatness. The proof of the desingularization theorem leads to an algorithm which has been implemented in Maple and \textsc{Singular}. Ana María Bravo, Santiago Encinas, and Orlando Villamayor U., A simplified proof of desingularization and applications, Rev. Mat. Iberoamericana 21 (2005), no. 2, 349 -- 458. Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of higher-dimensional varieties A simplified proof of desingularization and applications

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 study some properties of linear systems and the locus of linear systems on a complex projective algebraic curve which is a covering of another curve. In section 1, we prove the irreducibility of the \(W^1_d(X)\) for all \(d\geq g-h+1\) on a curve \(X\) of genus \(g\) which is a double covering of a general curve \(C\) of genus \(h>0\). And this result is sharp. In the proof of theorem 1.1, we use the equivalence of the irreducibility of \(W_d^1(X)\) and the connectivity of \(W_d^1(X)\), if \(W^1_d(X)\) has the positive expected dimension and is non-singular in codimension one [cf. \textit{W. Fulton} and \textit{R. Lazarsfeld}, Acta Math. 146, 271-283 (1981; Zbl 0469.14018)]. We also use the so-called Castelnuvo-Severi inequality for a double covering \(X\) of genus \(g\) over a curve \(C\) of genus \(h\); every base-point-free \(g^1_n\) on \(X\) is a pull-back of a \(g^1_{n/2}\) on \(C\) for any \(n\leq g-2h\) [cf. \textit{R. D. M. Accola}, ``Topics in the theory of Riemann surfaces'', Lect. Notes Math. 1595 (1994; Zbl 0820.30002), chapter 3]. In section 2, we consider a problem of base-point-free pencils of certain degree on a \(k\)-gonal curve as well as on a curve which is a double covering of a genus two curve. In proving the main results of section 2, we use enumerative methods and computatios in \(H^*(C_\alpha, \mathbb{Q})\) of various sub-loci of the symmetric product \(C_\alpha\) of the given curve \(C\). Specifically, we compare the fundamental class of \(C^1_\alpha: =\{D\in C_\alpha: \dim|D|\geq 1\}\) with the class of all irreducible components of \(C^1_\alpha\) whose general elements correspond to pencils on \(C\) with base points. This argument works because the latter components are all induced from the base curve of the covering and \(C^1_\alpha\) has the expected dimension. Throughout, we work over the field of complex numbers. double covering curves; linear systems; base-point-free pencils E. Ballico and C. Keem, Variety of linear systems on double covering curves, J. Pure Appl. Algebra 128 (1998), no. 3, 213--224. Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Variety of linear systems on double covering 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 begins with a nice overview of the theory of Macaulay inverse systems. The authors then consider the situation of the inverse system of the ideal of a zero-dimensional scheme \(\mathfrak Z\) in \(\mathbb P^n\), beginning with one that is supported at a point and then passing to the general case. This represents the focus of the paper -- how to pass from the local inverse systems of the irreducible components of \(\mathfrak Z\) to the global inverse system. When \(\mathfrak Z\) is locally Gorenstein, the authors give conditions under which for a general element \(F\) of degree \(d\) that is apolar to \(\mathfrak Z\), one can recover \(\mathfrak Z\) from \(F\). As a consequence, they show that a natural upper bound for the Hilbert function of Gorenstein Artin quotients of the coordinate ring of \(\mathfrak Z\) is achieved for large socle degree. They give some consequences for linkage, and for the uniqueness (in some cases) of generalized additive decompositions of a homogeneous form into powers of linear forms. Many of the results and remarks are labelled with short descriptions for the convenience of the reader. Macaulay inverse system; regularity degree; globalization; zero-dimensional scheme; Gorenstein Artin ring; irreducible components; generalized additive decomposition Cho, Y. H.; Iarrobino, A., Inverse systems of zero dimensional schemes in \(\mathbb{P}^n\), J. Algebra, 366, 42-77, (2012) Projective techniques in algebraic geometry, Commutative rings of differential operators and their modules, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes) Inverse systems of zero-dimensional schemes in \(\mathbb P^n\)

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 \(X\) denote the space of \(n \times n\)-skew symmetric matrices over the complex field \(\mathbb{C}\) under the natural action of \(\operatorname{GL}_n(\mathbb{C})\). The orbit closure \(\overline{O_k}\) of this action of the set of matrices \(O_k\) of rank \(2k\) are the Pfaffian varieties of matrices of rank \(\leq 2k\). Let \(S = \mathbb{C}[X]\) and \(\mathfrak{m}\) its homogeneous maximal ideal with \(E\) the injective hull of the residue field \(S/\mathfrak{m}\). Let \(R_k\) denote the localization of the homogeneous coordinate ring of \(\overline{O_k}\) at the maximal ideal. The Lyubeznik numbers \(\lambda_{i,j}(R_k)\) are determined by the decomposition of the local cohomology functors \(H^i_{\mathfrak{m}}(H^{\dim X -j}_{\overline{O_k} }(S) )= \oplus E^{\lambda_{i,j}(R_k)}\). The author computes the precise values \(\lambda_{i,j}(R_k)\) in the previous decompositions by studying the \(\mathcal{D}_X\)-module structure of \(H^{\bullet}_{\overline{O_k} }(S)\) where \(\mathcal{D}_X\) is the Weyl algebra of differential operators on \(X\) with polynomial coefficients. The local cohomology of the polynomial ring is a holonomic (and thus finite length) \(\mathcal{D}\)-module, and the simple composition factors in the case of Pfaffian varieties are known by \textit{C. Raicu} and \textit{J. Weyman} [J. Lond. Math. Soc., II. Ser. 94, No. 3, 709--725 (2016; Zbl 1357.14060)]. The author of the present work expands their work by describing the filtrations of these modules in the category \(\operatorname{mod_{GL}} (\mathcal{D}_X)\) of \(\operatorname{GL}\)-equivariant coherent \(\mathcal{D}\)-modules on \(X\). There is also a discussion of the work of \textit{A. C. Lörincz} and \textit{C. Raicu} [``Iterated local cohomology groups and Lyubeznik numbers for determinantal rings'', Preprint, \url{arXiv:1805.08895}], where they compute Lyubeznik numbers for determinantal rings. Pfaffian ideals; Lyubeznik numbers Homological functors on modules of commutative rings (Tor, Ext, etc.), Determinantal varieties, Local cohomology and commutative rings Lyubeznik numbers for Pfaffian 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 We relate the (co)homological properties of two objects: square-free monomial ideals and real coordinate subspace arrangements. The interest in studying such arrangements comes from the facts that they provide examples of arbitrary torsion in the cohomology of the complement of the arrangement and the complements provide examples of manifolds with properties similar to toric varieties, and toric varieties as quotients. A comparison of our formula for monomial ideals with the Goresky-MacPherson Formula for the cohomology of the complement of a subspace arrangement leads to theorem 3.1. This result states that the \(i\)-dimensional cohomology of the complement of a real coordinate subspace arrangement is computed by the Betti numbers in the \(i\)-strand in the minimal free resolution of a certain square-free monomial ideal. In corollaries 3.3 and 3.4 we show how this reveals an equivalence of results, which on the one hand were proved for subspace arrangements by \textit{A. Björner} [in: First European Congress of Mathematics, Paris, 1992, Vol. I, Progr. Math. 119, 321--370 (1994; Zbl 0844.52008)] and on the other hand were recently proved for monomial ideals by \textit{J. Eagon} and \textit{V. Reiner} [J. Pure Appl. Algebra 130, 265--275 (1998; Zbl 0941.13016)] and \textit{N. Terai} [Generalization of Eagon-Reiner theorem and \(h\)-vectors of graded rings, preprint]. Gasharov, V., Peeva, I., Welker, V.: Coordinate subspace arrangements and monomial ideals. In: Hibi, T. (ed.) Computational commutative algebra and combinatorics. Adv. Stud. Pure Math. vol. 33, pp. 65-74. Mathematical Society of Japan, Tokyo (2001) Topology of real algebraic varieties, Configurations and arrangements of linear subspaces, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Syzygies, resolutions, complexes and commutative rings, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Coordinate subspace arrangements and monomial 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 Let \(V\) be a toric variety associated to a cone \(\sigma\) over a field \(k\), and \(G\) the minimal generator system of the semi-group defined by \(\sigma\). A \(G\)-desingularization of \(V\) corresponds to a regular subdivision of \(\sigma\) whose edges contain the elements of \(G\). -- If \(\dim(\sigma)=2\), \(G\)-desingularization exists, if \(\dim(\sigma)=\dim(V)=2\), it is the minimal desingularization of \(V\). In this note, there is an example with \(\dim(V)=4\) without any \(G\)- desingularization (example 4). The main result of this note is the existence and the effective construction of \(G\)-desingularization when \(\dim(V)=3\) (theorem 1). \(G\)- desingularization is not unique in general, but different \(G\)- desingularizations may be related by sequences of flops (theorem 2). At the end of this note, there is a geometric interpretation of \(G\) in any dimension: there is a bijective correspondence between elements of \(G\) and essential divisors of \(V\) (theorem 5). toric 3-fold; toric variety; minimal generator; desingularization Bouvier, Catherine; Gonzalez-Sprinberg, Gérard: G-désingularisations de variétés toriques. C. R. Acad. sci. Paris sér. I math. 315, No. 7, 817-820 (1992) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds G-désingularisations de variétés toriques. (G-desingularizations of toric 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 Let \(V\) and \(W\) be finite dimensional vector spaces over an algebraically closed field of characteristic zero and let \(M\) be either (i) space \({\text{Hom}}(V, W)\), or (ii) one of the spaces \({\text{S}}^2(V^*)\), \({\Lambda}^2(V^*)\), \(V^*\otimes V^*\). Let \(G={\text{SL}}(W)\times {\text{SL}}(V)\) in case (i) and \(G={\text{SL}}(V)\) in case (ii). For the natural action of \(G\) on the direct sum of \(d\) copies of \(M\), the authors determine the irreducible components of the null-cone and a value of \(d\) at which the number of these components stabilizes. They also describe the cases where this null-cone is defined by the polarizations of the invariants on \(M\). null-cone; representation; bilinear form; Hilbert--Mumford criterion Bürgin, M.; Draisma, J., The Hilbert null cone on tuples of matrices and bilinear forms, Mathematische Zeitschrift, 254, 785-809, (2006) Geometric invariant theory, Representation theory for linear algebraic groups, Quadratic and bilinear forms, inner products The Hilbert null-cone on tuples of matrices and bilinear forms

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 related to resolution of singularities. We present here examples which explain why many arguments and proofs work in special situations, say small dimension or zero characteristic, but fail in general. This exhibits in particular the delicacy of resolution of singularity for arbitrary excellent schemes. We shall concentrate here on the classical approach developed by Zariski, Abhyankar, Hironaka and several other mathematicians towards a constructive proof of resolution of singularities by a sequence of well chosen monoidal transformations. The basic idea is to construct sufficiently fine local invariants of singularities which determine the center of blowing up at each stage as the locus of points on the variety where the invariants take their maximal values and to measure the improvement the variety undergoes when passing to the blown up variety by comparing these and possibly further invariants before and after the blowup. -- At present there is no completely satisfying answer to this objective. One reason is the lack of conceptuality of the proposed and studied invariants, already apparent in characteristic 0, the other, which is partly a consequence of the first, that the known invariants sometimes behave badly under blowup in positive characteristic. We shall set up a catalogue of cautions one has to be aware of when using standard resolution invariants or searching new ones. resolution of singularities Hauser, H.: Seventeen obstacles for resolution of singularities. The Brieskorn anniversary (1998) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Seventeen obstacles for 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 Over a field of characteristic zero the geometry of orbit closures for equioriented \(A_n\) quiver was first studied by \textit{S. Abeasis} et al. [Math. Ann. 256, 401-418 (1981; Zbl 0477.14027)] where it was established that the orbit closures are normal, Cohen-Macaulay, and have rational singularities. This result was generalized to the case of a quiver \(A_n\) with an arbitrary orientation by \textit{G. Bobiński} and \textit{G. Zwara} [Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)]. In the paper under review orbit closures for the non-equioriented \(A_3\) quiver are investigated. Namely, a minimal free resolution of the defining ideal of an orbit closure is explicitly constructed, a description of a minimal set of generators of the defining ideal is obtained, a classification of orbits closures which are Gorenstein is established. Gorenstein orbit closures; Lascoux resolution; Cohen-Macaulay varieties; Dynkin quivers; geometry of orbit closures; Bott vanishing theorem Representations of quivers and partially ordered sets, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Determinantal varieties, Singularities in algebraic geometry, Actions of groups on commutative rings; invariant theory, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Resolutions of defining ideals of orbit closures for quivers of type \(A_3\).

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 0487.00004. For part I see Inst. Élie Cartan, Univ. Nancy I 3, 33-55 (1981; Zbl 0572.14002).] Let (X,x) be a germ of a reduced analytic space of (pure) dimension d. A collection of d natural numbers (polar multiplicities) \[ M^*_{X,x}=\{m_ x(X),\quad m_ x(P_ 1(X)),...,m_ x(P_{d- 1}(X))\} \] corresponds to it. Here \(P_ k(X)\) is the (local) polar variety in general position of codimension k for the germ X, \(m_ x\) the multiplicity in the point x. The local polar variety \(P_ k(X)\) of codimension k can be defined in the following way. Let \((X,x)\to ({\mathbb{C}}^ N,0)\) be an imbedding of the germ (X,x) into a complex linear space, p: (\({\mathbb{C}}^ N,0)\to ({\mathbb{C}}^{d-k+1},0)\) be the projection along a subspace L in general position of dimension \(N-d+k-1\). The polar variety \(P_ k(x)\) is the closure in X for the set of critical points of the restriction of the projection p to the set \(X^ 0\) of non-singular points of the germ X. It is either empty or an analytic supspace of pure codimension k in X. Its multiplicity in the point x is denoted by \(m_ x(P_ k(X))\). It does not depend on the space L along which the projection is realized if L is chosen in general position. We have \(X=P_ 0(X)\), i.e., \(m_ x(X)=m_ x(P_ 0(x))\). - Main result of the paper under review: Theorem: Let X be a reduced, complex-analytic space of pure dimension d, Y be a non-singular analytic subspace of the space X, \(0\in Y\). The following conditions are equivalent: (1) the pair \((X^ 0,Y)\) satisfies the Whitney conditions (a),(b) in 0 \((X^ 0\) is the set of non-singular points of the space X); (2) the collection of polar multiplicities \(M^*_{X,y} (y\in Y)\) is constant for all \(y\in Y\) of a neighbourhood of the point 0. Thus, the pair \((X^ 0,Y)\) does not satisfy the Whitney conditions in the point 0 if and only if the multiplicity of one of the local polar varieties \(P_ k(X)\) of general form is not constant in a neighbourhood of 0 on Y. The fact, that the collections of polar multiplicities \(M^*_{X,x}\) can be (in a certain sense) computed by topological methods while the immediate checking of fulfilment of the Whitney conditions requires analytical computations, accounts for the significance of this result. Whitney conditions; local polar variety of codimension k; imbedding; multiplicity; polar multiplicities Teissier, B., Variétés polaires II: multiplicités polaires, sections planes et conditions de Whitney, Lecture Notes in Math., 961, (1982) Singularities in algebraic geometry, Multiplicity theory and related topics, Complex singularities Variétés polaires. II: Multiplicités polaires, sections planes et conditions de Whitney. (Polar varieties. II: Polar multiplicities, plane sections and the Whitney conditions)

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 a local reduced analytic ring \((R,m)\) of Krull dimension \(d\) and using classical results by Samuel, Rees and Ramanujam provides a geometric way of compute the multiplicity of an \(m\)-primary ideal of \(R\). Furthermore, particularizing the used ideas to develop the above result to the case of non-singular complex surfaces, the author gives a new and geometric proof of a result by Mumford. multiplicity of ideals; non-singular complex surfaces Multiplicity theory and related topics, Singularities in algebraic geometry, Singularities of curves, local rings, Plane and space curves, Local complex singularities, Invariants of analytic local rings, Equisingularity (topological and analytic) Linear systems and multiplicity 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 Let \(A^{n}\) be an affine space over a field \(K\) of characteristic zero and \(GA_{n}\) the group of polynomial automorphisms of \(A^{n}.\) Let \(\mathbf{n} =(n_{1},\dots ,n_{q})\in\mathbb{N}^{q}.\) Define \(A^{\mathbf{n}}=A^{n_{1}} \oplus\dots \oplus A^{n_{q}}\) and denote by \(\mathbf{x}_{i}=(x_{i,j})_{j=1,\dots n_{i}}\) coordinates in \(A^{n_{i}}.\) The author defines two algebraic groups of infinite dimension: 1. \(U_{\mathbf{n}}\subset GA_{|\mathbf{n}|}\), the block-triangular polynomial translations of the form \[ \mathbf{x}_{i}\mapsto\mathbf{x} _{i}+\mathbf{h}(\mathbf{x}_{1},\dots ,\mathbf{x}_{i-1}), \] where \(\mathbf{h}_{i}\) are systems of polynomials, 2. \(B_{\mathbf{n}}\subset GA_{|\mathbf{n}|}\) the automorphisms of the form \[ \mathbf{x}_{i}\mapsto S_{i}(\mathbf{x}_{i})+\mathbf{h} (\mathbf{x}_{1},\dots ,\mathbf{x}_{i-1}), \] where \(S_{i}\in GL_{n_{i}}(K)\) and \(\mathbf{h}_{i}\) are systems of polynomials. The group \(B_{\mathbf{n}}\) is a generalization of known groups of automorphisms: the affine group of automorphisms (\(q=1, \mathbf{n=}n_{1}\)) and the triangular group (\(n_{i}=1, i=1,\dots ,q)\). The main results of the paper are descriptions of regular automorphisms of \(U_{\mathbf{n}}\) and \(B_{\mathbf{n}}\). triangular automorphism; algebraic group; automorphism group Bodnarchuk, Yu.V.: On automorphisms of block-triangular polynomial translation groups. J. pure appl. Algebra 137, 103-123 (1999) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Birational automorphisms, Cremona group and generalizations On automorphisms of block-triangular polynomial translation 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 Our object of study is a rational map \(\Psi:\mathbb{P}^{s-1}_k\dashrightarrow\mathbb{P}^{n-1}_k\) defined by homogeneous forms \(g_{1},\ldots,g_{n}\), of the same degree \(d\), in the homogeneous coordinate ring \(R=k[x_{1},\ldots,x_{s}]\) of \(\mathbb{P}_{k}^{s-1}\). Our goal is to relate properties of \(\Psi\), of the homogeneous coordinate ring \(A=k[g_{1},\ldots,g_{n}]\) of the variety parameterized by \(\Psi\), and of the Rees algebra \(\mathcal{R}(I)\), the bihomogeneous coordinate ring of the graph of \(\Psi\). For a regular map \(\Psi\), for instance, we prove that \(\mathcal{R}(I)\) satisfies Serre's condition \(R_{i}\), for some \(i>0\), if and only if \(A\) satisfies \(R_{i-1}\) and \(\Psi\) is birational onto its image. Thus, in particular, \(\Psi\) is birational onto its image if and only if \(\mathcal{R}(I)\) satisfies \(R_{1}\). Either condition has implications for the shape of the core, namely, \(\text{core}(I)\) is the multiplier ideal of \(I^{s}\) and \(\text{core}(I)=(x_{1},\ldots,x_{s})^{sd-s+1}.\) Conversely, for \(s=2\), either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of \(g_{1},\ldots,g_{n}\), we give an explicit method to reduce the nonbirational case to the birational one when \(s=2\). Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Multiplicity theory and related topics, Varieties and morphisms, Rational and birational maps, Multiplier ideals Blowups and fibers of morphisms

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 property to be Koszul for the coordinate ring of a projective scheme (a refinement of being generated in degree one and with quadric relations), has been studied by many authors. Since a projectively normal variety is Koszul iff its hyperplane sections are, it is quite of interest to have Koszulness criteria for finite sets of points in \(\mathbb{P}^n\). A result by Kempf gives that any set of \(\leq 2n\) points in linear general position in \(\mathbb{P}^n\) is Koszul; in this paper criteria for Koszulness of non-generic sets of points are given which depend only on linear spans of subsets and their incidences. Namely, main results are: 1) if a finite set \(S \subset\mathbb{P}^n\) is such that \(S = S_1\sqcup S_2\), \(S_1, S_2\) linearly independent and such that span\((S_1)\cap S_2 = S_1\cap \text{span}(S_2) = \emptyset\), then \(S\) is Koszul. 2) Let \(S \subset\mathbb{P}^n\) be a Koszul set of points. Let \(S'\subset S\) be such that \(S-S'\) imposes independent conditions to quadrics containing \(S'\). Then \(S'\) is Koszul. As a consequence of 2) we get that if \(S \subset \mathbb{P}^n\) is a Koszul set of points which imposes independent conditions to quadrics, then every subset of \(S\) is also Koszul. The proofs use results about conditions under which homomorphism of graded rings preserve Koszulness and Koszul families of ideals in graded algebras. points; Koszul rings; minimal resolution DOI: 10.1016/j.jalgebra.2005.06.026 Syzygies, resolutions, complexes and commutative rings, Homological functors on modules of commutative rings (Tor, Ext, etc.), Algebraic cycles Koszul configurations of points in projective 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 It is well known that in positive (or finite) characteristic additional problems arise in the comparison of the infinitesimal and local points of view. For example trying to ``exponentiate'' the derivation \({\mathcal D}\) of a Lie algebra \(L\) to a formal automorphism \(\varphi=1+t{\mathcal D}+t^2\varphi_2+\cdots\), you meet a problem because the usual formula \(\varphi_n= \frac{1}{n!} {\mathcal D}^n\) stops working for \(n=p\), where \(p\) is the characteristic of the ground field. One can make a cocycle for \(H^2(L,L)\): \[ \psi(x,y)= \sum_{i=1,\dots,p-1} \frac{1}{i!(p-i)!} [{\mathcal D}^ix,{\mathcal D}^{p-i}y], \] which is denoted \(\text{Sq }{\mathcal D}\) and called obstruction. The author makes careful considerations of these problems and of the whole situation with deformation in this context. Following Gerstenhaber, he distinguishes geometric rigidity and formal analytic rigidity, provides several criteria for a finite-dimensional Lie or associative algebra \(L\) to be rigid in that sense, and discusses properties of the obstruction subspace in \(H^2(L,L)\). His Theorem 2 shows the special role of the automorphism scheme \(\Aut(L)\) in this context. As an application the scheme theoretic description of deformations of the Jacobson-Witt algebras \(W_n\) is given. deformations; obstructions; Lie algebra; group scheme; geometric rigidity; formal analytic rigidity; automorphism scheme; deformations of the Jacobson-Witt algebras Skryabin, S, Group schemes and rigidity of algebras in characteristic zero, J. Pure Appl. Algebra, 105, 195-224, (1995) Cohomology of Lie (super)algebras, Formal methods and deformations in algebraic geometry, Group schemes, Deformations of associative rings Group schemes and rigidity of algebras 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 \(\mathcal S\) be a connected and locally 1-connected space, and let \(\mathcal M\subset \mathcal S\). A decorated \(SL_2(\mathbb C)\)-\textit{local system} is an \(SL_2(\mathbb C)\)-local system on \(\mathcal S\), together with a chosen element of the stalk at each component of \(\mathcal M\). We study the decorated \(SL_2(\mathbb C)\)-\textit{character algebra} of \((\mathcal S,\mathcal M)\): the algebra of polynomial invariants of decorated \(SL_2(\mathbb C)\)-local systems on \((\mathcal S,\mathcal M)\). The character algebra is presented explicitly. The character algebra is shown to correspond to the \(\mathbb C\)-algebra spanned by collections of oriented curves in \(\mathcal S\) modulo local topological rules. As an intermediate step, we obtain an invariant-theory result of independent interest: a presentation of the algebra of \(SL_2(\mathbb C)\)-invariant functions on End\((\mathbb V)^m \oplus \mathbb V^n\), where \(\mathbb V\) is the tautological representation of \(SL_2(\mathbb C)\). local systems; rings of invariants; mixed invariants; mixed concomitants; skein algebra; cluster algebra; quantum cluster algebra; quantum torus; triangulation of surfaces DOI: 10.2140/agt.2013.13.2429 Actions of groups on commutative rings; invariant theory, Algebraic moduli problems, moduli of vector bundles, Topological methods in group theory, Invariants of knots and \(3\)-manifolds Character algebras of decorated \(SL_{2}(\mathbb C)\)-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 main goal of this paper is to provide new basepoint-free theorems using the framework of b-divisors and saturation of linear systems (introduced by Shokurov). The reformulation in these terms is very flexible and leads to several applications. In particular, a remarkable result is the generalization of a result of \textit{S. Fukuda} (Proposition 3.3 in [Int. J. Math. Sci. 30, No. 9, 521--531 (2002; Zbl 1058.14027)]) stated in Theorem 1.1. basepoint-freeness; b-divisors; saturation Fujino, O.: Base point free theorems--saturation, B-divisors and canonical bundle formula. Algebra Number Theory, to appear. arXiv:math/0508554v3 Divisors, linear systems, invertible sheaves, Adjunction problems, Minimal model program (Mori theory, extremal rays) Basepoint-free theorems: saturation, b-divisors, and canonical bundle 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 k be an algebraically closed field, H a connected linear algebraic group defined over k, and Y an affine k-variety on which H acts. Let R denote the coordinate ring of Y. This paper is concerned with the question of when the ring A of H-invariant elements of R is a finitely generated k-algebra. The prototypical case is when Y is a vector space on which H is represented by a representation \(\rho\). In this case the ring A is graded and \(I(\rho\),m) denotes the R-ideal generated by the m-th graded piece of A. The sum of these ideals is called the base locus ideal of \(\rho\). The base locus is stable if for all n sufficiently large \(I(\rho\),mn) and \(I(\rho\),m)n define the same sheaf of ideals on the projective space P(Y). For general affine Y, similar notions are defined via an H-equivariant embedding in a linear representation. The stability of the base locus is related to the finite generation of A. This is a complicated problem in general, and the author's results require a number of technical definitions to state properly, but the following theorem gives their flavor: In the case that Y is a vector space with linear H- action, if H has no characters and the base locus ideal is principal (hence stable) then A is finitely generated. action of algebraic group; finitely generation of ring of invariant elements Group actions on varieties or schemes (quotients), Commutative rings and modules of finite generation or presentation; number of generators, Geometric invariant theory, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Stable base loci of representations of algebraic 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 The aim of this paper is to generalize some results obtained previously by A. S. Merkur'ev and A. A. Suslin, concerning the structure of the \(p\)- component \(_p \text{Br} (F)\) of the Brauer group of a field \(F\) [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR Ser. Mat. 46, No. 5, 1011- 1046 (1982; Zbl 0525.18008) and \textit{A. S. Merkur'ev}, Math. USSR, Izv. 27, 141-157 (1986); translation from Izv. Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 4, 828-846 (1985; Zbl 0591.12026)]. This generalization deals with regular semilocal rings of geometric type, i.e. regular semilocal rings obtained by localisation of a finite type algebra over an infinite field at a finite set of prime ideals. For any regular semilocal ring of geometric type in which the prime number \(p\) is invertible and for every positive integer \(n\) the Kummer extension \(R_n\) of \(R\) is constructed, which is also a semilocal ring of geometric type. The Kummer extension \(R \hookrightarrow R_n\) has a standard algebra generator \(\xi\), which is the root in \(R_n\) of the \(p^n\)-th cyclotomic polynomial \(\Phi_{p^n} (t) \in R[t]\). Then, for \(a,b \in R^*_n\) the cyclic \(R_n\)-algebra \(A_\xi (a,b)\) is constructed, such that it is generated by two elements, which is a free \(R_n\)-module of rank \(p^{2n}\) and the complete table of multiplication of the linear basis is given. The first important result in order to generalize the Merkur'ev-Suslin result is that there exist an injective homomorphism: \(K_2 (R_n) \to {}_{p^n} \text{Br} (R_n) \otimes \Theta\) acting according the rule \(\{a,b\} \mapsto A_\xi (a,b) \otimes \xi\), where \(a,b \in R^*_n\) and where \(\Theta\) is the cyclic multiplicative subgroup of \(R_n\) generated by \(\xi\). Then, the subgroup \(_{p^n} K_2 (R_n)\) consists of the elements \(\{\zeta, a\}\) where \(\zeta \in \Theta\) and \(a \in R^*_n\). -- The above result is founded on the exact sequence of D. Quillen which characterizes \(K_2 (A)\) for regular semilocal rings \(A\) of geometric type. The second part of the paper presents interesting results concerning the generators of \(\text{Br} (R_n)\) and generators and relations of the \(p\)-primary component of \(\text{Br} (R_n)\) (theorems (5.1) and (6.1)). \(p\)-component of the Brauer group; regular semilocal ring of geometric type; Kummer extension Brauer groups of schemes, Regular local rings, \(K_2\) and the Brauer group On the structure of the Brauer group of a semilocal ring of geometric 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 The author demonstrates the following results: Let \(A\) be the ring of formal polynomials in \(n>1\) variables over an algebraically closed field \(k\), and \(M\) a torsion-free \(A\)-module whose rank be \(r>1)\). Then there exists a monomorphism \(i\colon M\to A^r\) such that \(i(M)\) includes some elements whose set of zeros over \(k\) has \(\text{cdm} >1\) in \(k^n\). This property is true also if \(A\) is any unique factorization domain finitely generated over \(A\), which allows one to affirm that: any coherent torsion-free algebraic sheaf \(\mathcal M^{(r)}\), of rank \(r>1\), over an affine variety with unique factorization algebra variety \((V, \mathcal A_V\) of dimension \(n>1\), is isomorphic to a subsheaf \(\mathcal M'^{(r)}\) of \(\mathcal A_V^r\) which has some sections whose set of zeros has \(\text{cdm} >1\) in \(V\). From this theorem it follows that for any coherent torsion-free algebraic sheaf \(\mathcal M^{(r)}\) over \((V, \mathcal A_V\) there exist \(r-1\) short exact sequences: \[ 0\longrightarrow \mathcal A_V^h\longrightarrow \mathcal M^{(r)} \longrightarrow \mathcal M^{(r-h)} \longrightarrow 0 \] \((1\le h\le r-1)\) of torsion-free sheaves, such that, for all \(h\), it is \[ \text{Supp}(\mathcal A_V^r)/ \mathcal M'^{(r)}) \subseteq \text{Supp}(\mathcal A_V^{(r-h)}/\mathcal M'^{(r-h)}). \] algebraic geometry S. Baldassarri-Ghezzo, ?Proprietà di fasci algebrici coerenti e lisci su varietà algebriche affini ad algebra fattoriale,? Rend. Semin. Mat. Univ. Padova, 1968,41, 12?30 (1969). Algebraic geometry Proprietà di fasci algebrici coerenti e lisci su varietà algebriche ad algebra fattoriale

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 morphism \(A\to B\) of noetherian commutative rings is regular iff it is a filtered inductive limit of smooth morphisms of finite type. This theorem allow to reduce the solvability in B of some polynomial equations over A to the solvability of some polynomial equations for which it is possible to apply the Implicit Function Theorem. The sufficiency is easy but the necessity is difficult even under some conditions of separability as it is given in this paper. The technique of our ''desingularization'' is mainly contained in the present paper the complete proof being given in our paper ''General Néron desingularization and approximation'' (submitted to Nagoya Math. J.) which contains also some applications of the above theorem. A stronger result (based on our theorem) is given by \textit{M. Cipu} and the author in Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984), which extends some theorems of \textit{M. Artin} and \textit{J. Denef} from Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, vol. II: Geometry, Prog. Math. 36, 5-31 (1983; Zbl 0555.14002). Néron desingularization; Zarèski uniformization theorem; smoothification along a section; regular morphisms; morphism of noetherian commutative rings; solvability of polynomial equations Popescu, D, General Néron desingularization, Nagoya Math. J., 100, 97-126, (1985) Global theory and resolution of singularities (algebro-geometric aspects), Commutative Noetherian rings and modules General Néron 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 In 1986 Eisenbud and Harris developed a theory to study the degeneration of a linear series on a family smooth curves as the curves degenerate to a certain type of reducible curve, the so-called compact type curve; all curves being defined over a base field of characteristic zero. Since then, some generalizations of this theory have appeared [see e.g. \textit{E.\ Esteves}, Mat. Contemp. 14, 21-35 (1998; Zbl 0928.14007) and \textit{M. Teixidor i Bigas}, Duke Math. J. 62, No. 2, 385--400 (1991; Zbl 0739.14006)]. In the present paper, the author proposes another generalization, removing the assumption on the characteristic of the base field and working with a more general family of curves, which he calls a smoothing family. This approach is more functorial in nature, and seems to be better suited to generalizations to higher-dimensional varieties and higher-rank vector bundles. In short, the author associates a functor to a smoothing family and the main result of paper states that this functor is representable by a projective scheme. The author also presents results on smoothing linear series from the special fiber when the dimension is expected, including the cases of positive and mixed characteristic. limit linear series; deformations Osserman, B.: A limit linear series moduli scheme. Ann. Inst. Fourier \textbf{56}(4), 1165-1205 (2006a) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic) A limit linear series moduli scheme

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 study configurations of points in projective space (over an algebrically closed field) which lie on rational normal curves. Namely, when considering sets of \(n\) ordered points in \({\mathbb P}^d\) that lie on a rational normal curve, we get that they give a subset of \(({\mathbb P}^d)^n\), and by taking its closure in the Zariski topology, we obtain a variety that is called its Veronese compactification: \(V_{d,n} \subset ({\mathbb P}^d)^n\). Any point of \(V_{d,n}\) parameterizes a configuration of \(n\) (possibly non-distinct) points supported on a flat limit of a rational normal curve. What the authors wants is to find equations for \(V_{d,n}\). We know that \(V_{d,n} = ({\mathbb P}^d)^n\) whenever \(d=1\) or \(n\leq d+3\), hence we can assume \(d\geq 2\) and \(n\geq d+4\); \(V_{d,n}\) is shown here to be irreducible and of dimension \(d^2 +2d +n-3\). When \(d=2\), a quite complete description of the equations for \(V_{2,n}\) can be given; namely what is shown is that: \begin{itemize} \item[--] \(V_{2,n}\) is defined scheme-theoretically by \({(\frac{n}{6})}\) determinants of certain \(6\times 6\) matrices of quadratic monomials (hence by degree \(12\) equations). \item[--] Let \(A = \{1,\dots, {(\frac{n}{6})}\}\) be a set of indices for those determinants; a subset of those determinants indixed by \({\mathcal T}\subset A\) defines \(V_{2,n}\) set-theoretically if and only if for all partitions \(I_1\cup \dots \cup I_6 = \{1,\dots,n \}\), it \(\exists {\mathcal J}\subset {\mathcal T}\) which meets every set \(I_j\) of the partition in one element. Consequently, at least \(\frac{2}{n-4}{(\frac{n}{6})}\) of those determinants are needed to define \(V_{2,n}\) set-theoretically. \item[--] \(V_{2,n}\) is Cohen-Macaulay, normal and it is Gorenstein if only if \(n=6\). \end{itemize} The case \(d\geq 3\) reveals to be far more complex; what is shown here is that, via the Gale transform, equations for \(V_{d,d+4}\) can be obtained from the ones of \(V_{2,d+4}\). Such equations, via the forgetful maps, give equations for any \(V_{d,n}\) which define a variety \(W_{d,n} \supseteq V_{d,n}\), and it is conjectured that \(W_{d,n} = V_{d,n}\cup Y_{d,n}\), where \(Y_{d,n}\) is the locus of degenerate points configurations (its equations are known). The inclusion \(W_{d,n} \supseteq V_{d,n}\cup Y_{d,n}\) is proved here for all \(d,n\), while the equality is obtained set-theoretically for \(d=3\) and \(n=d+4\). Other partial results are given for \(d=3,4\) and their study shows how the situation gets quite complex. parameter space; point configuration; gale transform; rational normal curve Generalizations (algebraic spaces, stacks) Equations for point configurations to Lie on a rational normal 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 Let \(C\) be a projective, connected, nodal algebraic curve of arithmetic genus greater than one, defined over an algebraically closed field \(k\) of characteristic zero, and let \(C_1, \dots, C_t\) be its components. Let \(S\) be the spectrum of a discrete valuation ring with residue field \(k\); a regular smoothing of \(C\) is a flat, projective map \(f: \mathcal{X} \rightarrow S\) whose generic fiber is smooth, whose special fiber is isomorphic to \(C\) and such that the total space \(\mathcal{X}\) is regular. Let \(K\) be the canonical sheaf on \(C\) and \(\beta\) a positive integer, then \(H^0(C, K^{\otimes \beta})\) defines the so-called \(\beta\)-canonical system; consider on the generic fiber of \(f\) the Weierstrass divisor of the \(\beta\)-canonical system and let \(W\) be the fundamental cycle on \(C\) associated to the subscheme that is the limit divisor. In this paper the authors describe \(W\) as a Weil divisor, under certain hypothesis (among them, \(t > 1\), \(C_i\) has positive genus for all \(i =1, \dots, t\) and hypothesis on the intersection of the components of \(C\)); conversely, they also find conditions for a Weil divisor to be a fundamental cycle of a limit Weierstrass divisor. limit of ramification points; nodal curves Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences Limit Weierstrass points on nodal reducible 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 Let \(V_0,V_1\) be two finite-dimensional vector spaces over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(L=\Hom (V_0,V_1) \times\Hom (V_1,V_0)\) and let \(H=\text{GL}(V_0) \times\text{GL}(V_1)\). Then \(H\) acts on \(L\) by \((g_1,g_2) \cdot(f_1,f_2) =(g_2f_1g_1^{-1}, g_1f_2g_2^{-1})\). An element \(f=(f_1,f_2)\in L\) is called a circular complex if \(f_1\circ f_2= f_2 \circ f_1=0\). The main theorem of the present paper asserts that the \(H\)-orbit closure \(\overline O_f\) of a circular complex \(f\) is normal, Cohen-Macaulay with rational singularities. It may be recalled that Strickland proved that each component of the full variety of the circular complexes in \(L\) is normal and Cohen-Macaulay. circular complexes; \(F\)-splitting; characteristic \(p\); Cohen-Macaulay variety V. B. Mehta, V. Trivedi, The variety of circular complexes and F -splitting, Invent. Math. 137 (1999), no. 2, 449--460. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Group actions on varieties or schemes (quotients), Singularities of surfaces or higher-dimensional varieties, Finite ground fields in algebraic geometry The variety of circular complexes and \(F\)-splitting

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 integer matrix \(A \in \mathbb{Z}^{d \times n}\), we study the natural mixed Hodge module structure in the sense of Saito on the Gauß-Manin system attached to the monomial map \(h : (C^\ast)^d \rightarrow \mathbb{C}^n\) induced by \(A\). We completely determine in the normal case the associated graded object to the weight filtration, by computing the intersection complexes with respective multiplicities that form its constituents. Our results show that these data are purely combinatorial, and not arithmetic, in the sense that they only depend on the polyhedral structure of the cone of \(A\), but not on the semigroup itself. In particular, we extend results of de Cataldo, Migliorini and Mustaţǎ to the setting of torus embeddings and give a closed form for the failure of the Decomposition Theorem in our context. If \(A\) is homogeneous and if \(\beta \in \mathbb{C}^d\) is an integral but not strongly resonant parameter, we use a monodromic Fourier-Laplace transform to carry the mixed Hodge module structure from the Gauß-Manin system to the GKZ-system attached to \(A\) and \(\beta\). In case \(A\) is derived from a normal reflexive Gorenstein polytope \(P\), Batyrev and Stienstra related certain filtrations on the generic fiber of the GKZ-system to the mixed Hodge structure on the cohomology of a generic hyperplane section inside the projective toric variety induced by \(P\). Our formulæ, phrased in terms of intersection cohomology groups on induced relative toric varieties, provide the necessary correction terms to globalize their computation. In particular, we document that on the GKZ-system the weight filtration will differ from Batyrev's filtration-by-faces whenever \(P\) is not a simplex: the intersection complexes contributing to the weight filtration measure the failure of \(P\) to be a simplex. Irrespective of homogeneity, we obtain a purely combinatorial formula for the length of the Gauß-Manin system, and thus for the corresponding GKZ-system. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration. toric varieties; intersection cohomology; decomposition theorem; weight filtration; GKZ systems Weight filtrations on GKZ-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 This is a continuation of the author's expository notes [Sûgaku 38, No. 2, 97--115 (1986; Zbl 0622.20040)]. In the first part, he interprets the relations between his theory of systems of general weights and the corresponding transformation groups and singularities of surfaces in \(\mathbb C^3\). In the present notes, he introduces the generalized root systems and shows that the root lattice can be imbedded in \(K3\)-lattice as a sublattice and discusses the invariants of the Weyl groups of root systems. To a regular system of weights with minimal exponent \(\varepsilon >0\), there corresponds a finite root system by McKay correspondence (cf. first part). Here, for any regular system of weights, he defines a ``generalized'' root system, where the root lattice \(H\) is the homology group \(H_2(X_1,\mathbb Z)\) of the Milnor fiber \(X_1\) corresponding to the system of weights, and the set \(R\) of the roots is the subset of vanishing cycles of \(H\) which is a natural generalization of finite root systems for the case \(\varepsilon >0\). He poses a number of reasonable axioms for ``generalized'' root systems which in fact is a generalization of finite or infinite root systems, and discusses Dynkin diagrams, Coxeter transformations which are related to the work of Gabrielov, Arnold, Brieskorn etc. To analyze the root lattice (in particular for the case \(\varepsilon <0)\), he discusses the imbedding of the root lattice into the homology group of \(K3\)-surface (\(K3\)-lattice) as a sublattice, namely he deals with the compactification of the Milnor fiber \(X_1\) for each case \(\varepsilon >0\), \(\varepsilon =0\), \(\varepsilon =-1\) or \(\epsilon\leq -1\) and \(\varepsilon\) is the only negative exponent. Let H be the root lattice corresponding to a regular system of weights, the Weyl group acts on the \({\mathbb{C}}\)-vector space \(Hom_{{\mathbb{Z}}}(H,{\mathbb{C}})\). He introduces and expounds his work on the ring \(S^+\) of W-invariant functions on a certain domain in \(Hom_{\mathbb Z}(H,\mathbb C)\), e.g. Chevalley type theorem for \(S^+\) and flat structure of \(S^+\) for the case \(\varepsilon >0\) and \(\varepsilon =0\). Finally, the author proposes and discusses some open problems related to the present theme [cf. Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012); Adv. Stud. Pure Math. 8, 479--526 (1986; Zbl 0626.14028)]. Milnor lattice; generalized root systems; root lattice; imbedded in K3- lattice; invariants of the Weyl groups of root systems; regular system of weights; homology group of K3-surface; Chevalley type theorem Simple, semisimple, reductive (super)algebras, Special surfaces, Singularities of surfaces or higher-dimensional varieties, Compact complex surfaces, Infinite-dimensional Lie (super)algebras The theory of systems of general weights and related topics. II: Its effects on the theory of singularities, general Weyl groups and their invariants, etc

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 work introduces a new framework for understanding uniform behavior of singularity measures such as Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity, and F-rational signature, for ideals varying in families of rings. Namely, the author calls the combination of a ring map $R \rightarrow A$ with an ideal $I$ of $A$ an \textit{affine $I$-family} if $A/I$ is module-finite over $R$, $I \cap R = 0$, and certain dimension formulas hold. (This is a restricted version of Lipman's notion of an $I$-family from [\textit{J. Lipman}, Lect. Notes Pure Appl. Math. 68, 111--147 (1982; Zbl 0508.13013)]). Then one analyzes the ideals $I(\mathfrak p)$, $I(\mathfrak p)^{[p^e]}$ (when char $k(\mathfrak p) = p>0$) and $I(\mathfrak p)^n$ for $\mathfrak p \in $Spec$(R)$ and various $e, n \in \mathbb N$ in the rings $R(\mathfrak p)$, where the notation $(\mathfrak p)$ means to tensor over $R$ with the residue field $k(\mathfrak p)$ of $\mathfrak p$. This framework allows the author to recover Lipman's result [loc. cit.] on upper semicontinuity of Hilbert-Samuel multiplicity on the prime spectrum, by showing that the \textit{terms} defining Hilbert-Samuel multiplicity as a limit are also upper semicontinuous in the family. He also recovers some of his own results (see [Compos. Math. 152, No. 3, 477--488 (2016; Zbl 1370.13006)]) on semicontinuity of Hilbert-Kunz multiplicity on the prime spectrum, again by analyzing the terms, where in this case one has an affine $I$-family $R \rightarrow S$ with reduced fibers of dimension $=$ height$(I)$, where $R$ is F-finite. He further obtains upper semicontinuity of Hilbert-Kunz multiplicity in an affine family where char $R=0$ and the characteristics of the fibers can vary but all residue fields of $R$ are F-finite when they are positive characteristic. In particular, when $R = \mathbb Z$, this answers a question of Claudia Miller from [\textit{H. Brenner} et al., J. Algebra 372, 488--504 (2012; Zbl 1435.13014)] in pursuance of obtaining a sensible notion of Hilbert-Kunz multiplicity in equal characteristic zero. The author also parlays his methods to show that for local algebras essentially of finite type over a prime characteristic field, the infimum in Hochster and Yao's definition of F-rational signature is actually achieved. He thus recovers a special case of the result in [\textit{M. Hochster} and \textit{Y. Yao}, ``F-rational signature and drops in the Hilbert-Kunz multiplicity'', Preprint] that the F-rational signature of the ring is positive if and only if the ring is F-rational. multiplicity; Hilbert-Samuel polynomial; Hilbert-Kunz multiplicity; semicontinuity; families Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Singularities in algebraic geometry, Deformations of singularities, Fibrations, degenerations in algebraic geometry On semicontinuity of multiplicities in families

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 any variety in characteristic zero. Let \(V \subset X\) be an open subset that has toroidal singularities. We show the existence of a canonical desingularization of \(X\) except for V. It is a morphism \(f : Y \to X\), which does not modify the subset \(V\) and transforms \(X\) into a toroidal embedding \(Y\), with singularities extending those on \(V\). Moreover, the exceptional divisor has simple normal crossings on \(Y\). The theorem naturally generalizes the Hironaka canonical desingularization. It does not modify the nonsingular locus \(V\) and transforms \(X\) into a nonsingular variety \(Y\). The proof uses, in particular, the canonical desingularization of logarithmic varieties recently proved by Abramovich-Temkin-Włodarczyk. It also relies on the established here canonical functorial desingularization of locally toric varieties with an unmodified open toroidal subset. As an application, we show the existence of a toroidal equisingular compactification of toroidal varieties. All the results here can be linked to a simple functorial combinatorial desingularization algorithm developed in this paper. resolution of singularities, toroidal resolution, logarithmic resolution Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Compactifications; symmetric and spherical varieties Functorial resolution except for toroidal locus. Toroidal compactification

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 module \(M\) over a ring \(R\) of characteristic \(p > 0\), consider a collection \(C\) of \(p^{-e}\)-linear maps on \(M\) for various \(e > 0\) . By definition \(p^{-e}\)-linear maps are additive maps \(\phi : M \to M\) which satisfy \(\phi(r^{p^e} m) = r \phi(m)\) for \(r \in R\) and \(m \in M\). When these maps form a ring under composition, the author of this paper defines the ring to be a \textit{Cartier-algebra} (which we note is not an \(R\)-algebra since \(R\) is not generally central). In the special case that \(M = R\) and \(R\) is reduced, this notion was introduced by the reviewer in [Trans. Am. Math. Soc. 363, No. 11, 5925--5941 (2011; Zbl 1276.13008)] as a generalization of several types of pairs and triples in \(F\)-singularity theory. In the case that \(R\) is complete and one looks at all possible maps, this is the same as the ring of Frobenius operators on the injective hull of the residue field as studied by \textit{G. Lyubeznik} and \textit{K. E. Smith} [Trans. Am. Math. Soc. 353, No. 8, 3149--3180 (2001; Zbl 0977.13002)]. Complementarily, the case of a Cartier-algebra made up of powers of a single map has been studied extensively by the author of the paper under review and \textit{G. Böckle} [J. Reine Angew. Math. 661, 85--123 (2011; Zbl 1239.13007)]. Regardless, the author of this already influential paper goes much further in studying the structure of these Cartier algebras. Notably, he proves a version of the Hartshorne-Speiser-Lyubeznik-Gabber finiteness theorem in this context. He also develops an interesting theory of test submodules with respect to any Cartier-algebra (including for non-reduced rings). He shows that this test submodule exists if \(R\) is of finite type over a field. As a corollary of this work, he finds new criteria for discreteness and rationality of \(F\)-jumping numbers of test ideals in the more usual \(F\)-singularity setting of pairs and triples. Cartier module; \(p^{-1}\)-linear map; test ideal Blickle, M., Test ideals via \(p^{- e}\)-linear maps, J. Algebraic Geom., 22, 1, 49-83, (2013) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplier ideals, Singularities in algebraic geometry Test ideals via algebras of \(p^{-e}\)-linear 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 devoted to questions related to transversality theory. A bound is given for the dimension of the singular locus of the general element of a linear system \(\mathcal L\) inside and outside the base locus \(B\) of \(\mathcal L\). The method is algorithmic and can be implemented into a computer algebra system. Examples shows that the bound is sharp. A result of Speiser about the ``not too ramified'' morphisms is improved. Remark 3.4 shows the applicability of this paper to certain cases where a criterion of Zhang for the smoothness of the general element of \(\mathcal L\) is not usable. linear systems; dimension of the singular locus; singularity; linear system; algorithmic method; ramification Divisors, linear systems, invertible sheaves, Ramification problems in algebraic geometry, Computational aspects in algebraic geometry, Singularities in algebraic geometry Linear systems, singularity and not too ramified morphisms

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 and \(\mathrm{Var}/K\), the category of reduced schemes of finite type over \(K\). An algebraic ring over \(K\) (originally introduced by \textit{M. J. Greenberg} [Ann. Math. (2) 73, 624--648 (1961; Zbl 0115.39004)]) is defined to be a `ring object' \(A\) in \(\mathrm{Var}/K\) whose underlying scheme is affine (irreducibility of the underlying scheme is not required), i.e., \(A\) additionally comes with an addition map \(\alpha : A\times A\to A\) and a multiplication map \(\mu : A\times A\to A\) which give it the structure of an associative ring. Every finite dimensional associative \(K\)-algebra \(B\) naturally gives rise to an algebraic ring structure \(\tilde{B}\) on the underlying \(K\)-vector space of \(B\) by virtue of its addition and multiplication operations. Note that \(\tilde{B}(K) = B\). If \(\mathrm{char}(K)=0\), it was shown by \textit{M. J. Greenberg} [Trans. Am. Math. Soc. 111, 472--481 (1964; Zbl 0135.21503)] that all algebraic rings arise in this fashion. However, if \(\mathrm{char}(K)=p>0\), there exist algebraic rings which do not come from \(K\) algebras as shown by the motivating example of \(A = \mathrm{G}_a\oplus \mathrm{G}_a\) with the twisted multiplication given by \((x,y)\star (a,b) = (xa, x^pb+a^py)\). However, the algebra of dual numbers \(B=K[t]/(t^2)\) gives rise to a closely related algebraic ring \(\tilde{B}=\mathbb{G}_a\oplus \mathbb{G}_a\) with the multiplication given by \((x,y)\cdot (a,b)=(xa, xb+ay)\), the connection given by the algebraic ring homomorphism \(\psi : \tilde{B}\to A\) sending \((x,y)\mapsto (x,y^p)\) which induces an isomorphism of abstract rings \(\tilde{B}(K) = B\to A(K)\) at the \(K\)-points level. The authors show that this phenomenon holds true more generally (after imposing some mild conditions on \(A\)), namely that every algebraic ring is closely related to a finite-dimensional associative \(K\) algebra up to an inseparable isogeny (Theorem 1.1). The proof (given in Section 3) proceeds by investigating the structure of commutative \textit{perfect} algebraic rings in char \(p\) and using Greenberg's \textit{perfectization} functor to study the general case. As an application, rigidity questions for representations of Chevalley groups over rings are studied. Let \(\phi\) be a reduced, irreducible root system of rank at least \(2\) and \(G = G(\phi)\), the associated \textit{universal Chevalley Demazure group scheme} defined over \(\mathbb{Z}\). For every root \(\alpha\in \phi\), there is a canonical morphism \(e_{\alpha} : \mathbb{G}_a\to G\) which induces a group homomorphism \(e_{\alpha}: (R, +)\to G(R)\) for every commutative ring \(R\). The subgroup generated by \(e_{\alpha}(R)\) for all roots \(\alpha\in \phi\) is called the \textit{elementary subgroup} \(E(\phi, R)\) of \(G(R)\). In [the second author, Proc. Lond. Math. Soc. (3) 102, No. 5, 951--983 (2011; Zbl 1232.20049)], abstract representations \(\rho : E(\phi, R)\to \mathrm{GL}_n(K)\) were analyzed when \(\mathrm{char}(K)=0\). In this paper, the same is studied in the positive characteristic setting, leading to Theorem (1.2) which shows that under mild assumptions on \(R\), every abstract finite dimensional representation \(\rho : E(\phi, R)\to \mathrm{GL}_n(K)\) can be essentially factored as follows: There exists a commutative finite dimensional \(K\) algebra \(B\), a ring homomorphism \(f : R \to B\) with Zariski-dense image, and a morphism of algebraic groups \(\sigma : G(B)\to \overline{\rho(E(\phi, R))}\) such that for a suitable finite index subgroup \(\Gamma\) of \(E(\phi,R)\) we have \(\rho|_{\Gamma} = (\sigma\circ F)|_{\Gamma}\) where \(F : E(\phi, R) \to G(B)\) is the group homomorphism induced by \(f\). As a further application of this theorem, the authors give a different proof to recover the following result of Seitz (Proposition 4.3), namely: Given an infinite perfect field \(k\) of characteristic \(p > 0\), \(K\) any algebraically closed field of char \(p\), \(\phi\) a reduced irreducible root system of rank at least \(2\), \(G=G(\phi)\) its universal Chevalley Demazure group scheme, \(Y\) an algebraic group over \(K\), and \(\rho : G(k)\to Y (K)\) a nontrivial abstract homomorphism, then \(\rho\) can be factored as \(G( k ) \to^{\sigma} G ( K ) \times \dots \times G ( K ) \to^{\psi} Y ( K )\) where \(\psi\) is a morphism of algebraic groups and \(\sigma\) is a \textit{twisted diagonal embedding}, i.e. each component \(G(k)\to G(K)\) of \(\sigma\) arises from a field homomorphism \(k\to K\). algebraic groups; representations; algebraic rings; positive characteristic Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups, Group schemes, Group varieties On abstract representations of the groups of rational points of algebraic groups 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 We study the Cox ring and monoid of effective divisor classes of \(\overline{M}_{0,n} \cong \text{Bl} \mathbb P^{n-3}\), over a ring \(R\). We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on \(\{1,\dots, n-1\}\) with weights in \(R \smallsetminus \{0 \}\) satisfying a zero-tension condition. This leads to a combinatorial criterion, satisfied by many triangulations of closed manifolds, for a divisor class to be among the minimal generators for the effective monoid. For classes obtained as the strict transform of quadrics, we present a complete classification of minimal generators, generalizing to all \(n\) the well-known Keel-Vermeire classes for \(n=6\). We use this classification to construct new divisors with interesting properties for all \(n \geq 7\). Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) A simplicial approach to effective divisors in \(\overline{M}_{0,n}\)

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 studies linear systems of curves defined by fixed points of given multiplicities on Hirzebruch surfaces \({{\mathbb F}_n}\). If \(\operatorname{Pic}{\mathbb {F}_n} = \langle H,F\rangle\), where \(F\) is a fiber and \(H^2=n\), \(F\cdot H=1\), the linear systems under consideration are of type \({\mathcal L}(a,b,m_1,\dots,m_s)\), i.e. the subsystem of the complete linear system \(| aF+bH|\) given by the curves having multiplicity at least \(m_i\) at \(P_i\), for \(s\) generic points \(P_1,\dots ,P_s\). In analogy with what happens in \({\mathbb P}^2\), the author conjectures that such linear systems are special (i.e. their dimension is bigger than expected) if and only if they are \((-1)\)-special, i.e. there is a \((-1)\)-curve \(E\) such that the base locus of \({\mathcal L}(a,b,m_1,\dots,m_s)\) contains \(\alpha \Gamma_n + tE\), \(t\geq 2\), \(\alpha \geq 0\). Here \(\Gamma_n\in | H-nF|\) is the curve with \(\Gamma ^2=-n\), while a \((-1)\)-curve \(E\) is an irreducible curve such that \(E^2=-1=K_{{\mathbb F}_n}\cdot E\). In the paper the conjecture is proved for \(m_1=\cdots =m_s\leq 3\); the main idea of the proof is to list all \((-1)\)-special systems with multiplicities \(m_i\leq 3\) via birational transformations \({\mathbb F}_n \rightarrow {\mathbb F}_{n-1}\) (a generalization of quadratic transformations in \({\mathbb P}^2\)) and then to work with suitable deformations of \({\mathbb F}_n\). fat points; Hirzebruch surfaces; \((-1)\)-special A. Laface: ''On linear systems of curves on rational scrolls'', Geom. Dedicata, Vol. 90, (2002), pp. 127--144. Divisors, linear systems, invertible sheaves, Birational automorphisms, Cremona group and generalizations, Rational and ruled surfaces On linear systems of curves on rational 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 Let \(S\) be a scheme of characteristic \(p>0.\) Let \(X\) be a \(p\)-divisible group over \(S\). A filtration \(0=X_{0}\subset X_{1}\subset \cdots \subset X_{m}=X\) with each \(X_{i}\) a \(p\)-divisible group is called a slope filtration of \(X\) provided there exists a strictly decreasing set of rational numbers \(\lambda _{1},\lambda _{2},\ldots ,\lambda _{m}\) in \([0,1]\) such that for all \(i>1\) we have \(X_{i}/X_{i-1}\) is isoclinic of slope \(\lambda _{i}.\) Such a filtration may not exist, although it does in the case where \(S \) is a field [see \textit{T. Zink}, Duke Math. J. 109, 79-95 (2001; Zbl 1061.14045)]. If \(X\) satisfies further conditions involving powers of the Frobenius, then \(X\) is completely slope divisible. The issue in this paper is to investigate when a certain class of \(p\)-divisible groups admits a slope filtration. Here the \(p\)-divisible groups that are considered have constant Newton polygon. An example is given to show that even in this case a slope filtration may not exist. However, the main result here is the following (from the paper): ``Let \(S\) be an integral, normal Noetherian scheme. Let \(X\) be a \(p\)-divisible group over \(S\) of height \(h\) with constant Newton polygon. Then there is a completely slope divisible \(p\)-divisible group \(Y\) over \(S\), and an isogeny: \(\varphi :X\rightarrow Y\) over \(S\) with \(\deg \varphi \leq N(h)\)''. A corollary to this result is that every \(p\)-divisible group with constant Newton polygon over a normal base has, up to isogeny, a slope filtration. From this theorem, the authors are able to prove the following. Let \(R\) be a Henselian local ring with residue field \(k\), and let \(h\in \mathbb{N.}\) Let \( S=\text{Spec}\left( R\right)\), and let \(X\) and \(Y\) be isoclinic \(p\)-divisible groups over \(S\) with ht\(\left( X\right)\), ht\(\left( Y\right) <h.\) Then there exists a \(c\) such that, given any homomorphism \(\psi :X_{k}\rightarrow Y_{k},\) the homomorphism \(p^{c}\psi \) lifts to a homomorphism \(X\rightarrow Y.\) From this it follows that the category of isoclinic \(p\)-divisible groups up to isogeny over \(R\) is equivalent to the same category over \(k.\) Finally, examples are given. The first example is the one referred to above -- a \(p\)-divisible group with constant Newton polygon which does not have a slope filtration. The other example given in the case \(S\) is not normal is of an \(X\) which is not isogenous to a \(Y\) admitting a slope filtration. These illustrate the necessity of the conditions on \(S\) as well as the ``up to isogeny'' portion of the corollary. \(p\)-divisible groups; characteristic \(p\); slope filtrations; Newton polygon Oort, Frans; Zink, Thomas, Families of {\(p\)}-divisible groups with constant {N}ewton polygon, Documenta Mathematica, 7, 183-201, (2002) Formal groups, \(p\)-divisible groups, \(p\)-adic cohomology, crystalline cohomology, Toric varieties, Newton polyhedra, Okounkov bodies Families of \(p\)-divisible groups with constant Newton polygon

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 normal variety of dimension at least \(3\). Let \(\mathcal M\) be a linear system on \(X\) with no fixed components and let \(O \in X\) be a smooth point such that \(O\) is a center of canonical singularities of the log-pair \((X,\frac{1}{n}{\mathcal M})\) for some \(n\in {\mathbb N}\). Assume that \((X,\frac{1}{n}{\mathcal M})\) is not log-terminal at~\(O\). According to a well-known result of Pukhlikov, the intersection multiplicity at \(O\) of two general divisors from \(\mathcal M\) is at least \(4 n^2\). This result, together with the Noether-Fano-Iskowskikh inequality, implies the non-rationality of some Fano manifolds, and it is sharp if \(X\) has dimension~\(3\). Moreover, similar results hold when \(O\) is a singular point. Assume now that \(\dim X\geq 4\), \(S_1, S_2\) are general divisors of \(\mathcal M\), \(O \in X\) is a smooth point (resp. an isolated ordinary double point) such that \(O\) is a center of canonical singularities of \((X,\frac{1}{n}{\mathcal M})\) and suppose that the singularities of \((X,\frac{1}{n}{\mathcal M})\) are canonical outside \(O\). Denote by \(\pi\) the blow-up of \(O\) and by \(E\) the exceptional divisor of \(\pi\). In the paper under review, the author proves that there exists a linear subspace \(\Lambda \subset E\) of codimension \(2\) (resp. a linear subspace \(\Lambda \subset {\mathbb P}^r\) of codimension~\(3\) with \(\Lambda \subset E\)) such that \[ \text{mult}_{O} (S_1 \cdot S_2 \cdot \Delta)\geq 8 n^2 \quad (\text{resp. } \geq 6 n^2) \] for every effective divisor \(\Delta\) on \(X\) such that \(O\) is a smooth point (resp. an ordinary double point) of \(\Delta\), \(\pi^{-1}(\Delta)\) contains \(\Lambda\) and \(\Delta\) contains no subvarieties of \(X\) of codimension \(2\) that are contained in the base locus of \(\mathcal M\). In particular, when \(O\) is an isolated ordinary double point and \(\dim X \geq 6\), by Lefschetz, he derives \(\text{mult}_{O} (S_1 \cdot S_2) > 6 n^2\). Moreover, these results imply the birational superrigidity and the non-rationality of many higher-dimensional Fano varieties of degree \(6\) and \(8\). Cheltsov, IA, Local inequalities and the birational superrigidity of Fano varieties, Izvestiya: Mathematics, 70, 605-639, (2006) Rational and birational maps, Fano varieties Local inequalities and birational superrigidity of Fano 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 From the author's abstract: ``The aim of this paper is to revise the theory of clusters on infinitely near points for arbitrary fields.'' Let \(K\) be a field, and let \(\Omega(K)\) be the set of two-dimensional regular local rings which have \(K\) as field of quotients; for \(R\in\Omega(K)\) the rings \(S\in\Omega(K)\) with \(S\supset R\) are said to be infinitely near to \(R\). Let \(R\in \Omega(K)\) and \(\mathfrak m\) be the maximal ideal of \(R\); every homogeneous principal prime ideal \(p\in \text{gr}_{\mathfrak m}(R)\) determines \(S_p\in \Omega(K)\) with \(S_p\supset R\); \(S_p\) is said to be a quadratic transform of \(R\), and the family of quadratic transforms of \(R\) is said to be the rings in the first neighborhood \(N_1(R)\) of \(R\) (one sets \(N_0(R)=\{R\}\)). Let \(R\subsetneqq S\) lie in \(\Omega(K)\); then there exists a uniquely determined sequence \(R=R_0\subset R_1\subset\cdots\subset R_h=S\) where, for \(i\in\{1,\dots,s\}\), \(R_i\) is a quadratic transform of \(R_{i-1}\); this sequence is called the quadratic sequence from \(R\) to \(S\). Now \(R\) determines an ideal in \(S\), the exceptional divisor; in Lemma 2.17 the author proves that the result which is well-known in the classical case -- where \(R\) is essentially of finite type over an algebraically closed field -- holds also in this more general setting. The order function \(\text{ord}_R\) of the maximal ideal of \(R\) determines a discrete valuation ring \(V\) of \(K\); \(S\supsetneqq R\) is proximate to \(R\), \(S\succ R\), if \(S\subset V\). Let \(p\) be as above, and let \(n_p\) be the order function of \(p\,\text{gr}_{\mathfrak m}(R)\). Then \(\nu_p(-)=(\text{ord}_R(-),(n_p(-))\) defines a valuation of \(R\). In Prop.\ 2.27 the author determines \(\nu_p(f)\) for \(f\in R\setminus\{0\}\) in terms of the points infinitely near to \(R\); a similar result for finite-colength ideals of \(R\) was shown by \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, Vol. I, 203--231 (1988; Zbl 0693.13011), Prop.\ 2.3 and Lemma 2.4]. In section 3 the author introduces the notion of a cluster \(\mathcal C\) (for the notion of a cluster in the classical case (cf.\, e.g., [\textit{A. Casas-Alvero}, Singularities of Plane Curves. London Mathematical Society Lecture Note Series. 276. Cambridge: Cambridge University Press (2000; Zbl 0967.14018)]), and defines, following Lipman [loc.~cit.], the refined proximity matrix \(P'_{\mathcal C}\) [there is a typo in the definition: if \(S\prec T\), then \(p'_{S,T}=[T:S]\)] and the total proximity matrix \(\widetilde R_{\mathcal C}\) [\,there is a typo: if \(S\prec T\), then \(\widetilde p_{S,T}=[T:R]\)]. Now let \[ X=X_s\overset{\pi_s}\rightarrow X_{s-1}\cdots X_1\overset{\pi_1}\rightarrow X_0\eqno(*) \] be a sequence of point blow-ups of two-dimensional regular schemes; more precisely, let \(x_0\in X_0\) be a closed point, let \(X_1=\text{Bl}_{x_0}(X_0)\) be the blow-up of \(X_0\) in \(x_0\), choose a closed point \(x_1\in X_1\) lying over \(x_0\), let \(X_2={\text{Bl}}_{x_1}(X_1)\) be the blow-up of \(X_1\) etc. Setting \(R_i:=\mathcal O_{X_i,x_i}\) for \(i\in\{0,\dots,s\}\), we get the quadratic sequence \[ R_0\subset R_1\subset\cdots\subset R_s; \] this sequence is called the cluster associated to \((*)\). In Theorem 4.8 the author determines the elements of the intersection matrix \(N_{\mathcal C}\) associated to this cluster [in all the formulae on pp. 453--454 one must replace \(\widetilde p_{U,S}\) by \(\widetilde p_{S,U}\) and \(p_{U,T}\) by \(p_{T,U}\)]. In section 5, the author defines the Hamburger-Noether tableau of a pair \((x,y)\) of elements in \(V=k[\![\,t\,]\!]\), \(k\) a field, following \textit{P. Russell} [Manuscr. Math. 31, 25--95 (1980; Zbl 0455.14018)] (there is no need to use the algebraic closure of \(k\); the elements \(a_i\) lie in \(k\)). In the last two sections the author works with theses tableaus. two-dimensional regular local ring; infinitely near point; point blowing-up; proximity; intersection matrix; Hamburger-Noether tableau; curvette Moyano-Fernández, J.J.: Curvettes and clusters of infinitely near points. To appear in Rev. Mat. Complut. doi: 10.1007/s13163-010-0048-1 Regular local rings, Singularities of curves, local rings Curvettes and clusters of infinitely near 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 author constructs explicitly some resolutions of singularities in many relevant examples that appear often in moduli problems. The base is an algebraically closed field of characteristic \(p\geq 2\), which is then generalized to \(\text{Spec}({\mathbb{Z}}_p )\). The main new idea is to use compactification of symmetric spaces and line bundles on flag varieties. In the first part of the paper, which has an independent interest in itself, it is shown that the compactification of symmetric spaces \(X=G/H\) given by \textit{C. De Concini} and \textit{C. Procesi} [Invariant theory, Proc. 1st 1982 Sess. C. I. M. E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] works in any characteristic. More precisely \(G\) is an adjoint semisimple Chevalley group and \(H\) is the invariant group of an involution of \(G\). This part relies on many other results in the literature, but it is almost self-contained and clearly written. If \(\lambda\) is a dominant weight, the compactification \(\overline{X}_{\lambda}\) is defined as the closure of \(X\) in a representation space \({\mathbb{P}} (V({\lambda}^*))\). There is a smooth \(\overline{X}\) which dominates the other \(\overline{X}_{\lambda}\). The author shows that there is a Frobenius splitting of the above compactification (i.e. a section of the injection \({\mathcal O}\to \text{Frob}_* ({\mathcal O})\) ). The splitting induces compatible splittings on all strata. The cohomology of line bundles on \(\overline X\) can be computed in many cases by using this splitting. It follows by these cohomological computations and by a criterion of \textit{G. Kempf} [``Toroidal embeddings. I'', Lect. Notes Math. 339, 41-52 (1973; Zbl 0271.14017)] that \(\overline {X}_{\lambda}^{norm}\) has rational singularities, improving previous results of several authors that showed that these singularities are Cohen-Macaulay. Then the author checks the projective normality of \(\overline{X}_{\lambda}\) in many cases, e.g. for \(X\) equal to \(G\times G\) quotiented by the diagonal. This machinery is applied to several examples of singularities. Additional informations about the strata at infinity are also obtained. A basic type of singularity studied in the examples is that of two \(n\times n\) matrices \(B\) and \(C\) satisfying \(B\cdot C=C\cdot B=p\text{ Id}\). These singularities appear in the author's paper [\textit{G. Faltings}, Math. Ann. 304, 489-515 (1996; Zbl 0847.14018)] about moduli-stacks for bundles on semistable curves. Some analogous examples for the other classical groups are also considered. resolutions of singularities; characteristic \(p\); compactification of symmetric spaces; moduli space; rational singularities; moduli-stacks for bundles on semistable curves Faltings, G.: Explicit resolution of local singularities of moduli-spaces. J. reine angew. Math. 483, 183-196 (1997) Global theory and resolution of singularities (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Singularities of curves, local rings Explicit resolution of local singularities of 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 Let \((R,\mathfrak{m})\) be a local domain essentially of finite type over a field. A regular alteration \(\pi:X\rightarrow \text{Spec} R\) is a proper and generically finite map with \(X\) a regular scheme. In this paper, the authors prove a characterization of regularity of the local ring \(R\) through alterations. They establish that for arbitrary characteristic, \(R\) is regular if and only if for every regular alteration \(\pi:X\rightarrow \text{Spec} R\), the derived image of the structure sheaf has finite projective dimension, i.e., \(\text{pd}_R\mathbf{R}\pi_*\mathscr{O}_X\leq \infty\). If \(R\) has characteristic zero, then \(R\) is regular if and only if for every regular alteration \(\pi:X\rightarrow \text{Spec} R\), \(\text{pd}_R\mathbf{R}\pi_*\mathscr{O}_X=0\). The proof of the main result is divided into positive characteristic and characteristic zero cases. The main contribution of the paper is the proof of characteristic zero case, for which the authors use multiplier ideals and multiplier submodules. multiplier ideals; projective dimension; regular rings; rational singularities Multiplier ideals, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Vanishing theorems in algebraic geometry, Homological dimension and commutative rings, Local cohomology and commutative rings, Singularities in algebraic geometry A Kunz-type characterization of regular rings via alterations

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 computable criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which \(p\)-kernels occur in an isogeny class of \(p\)-divisible groups over an algebraically closed field of positive characteristic. As an application we obtain a criterion for the non-emptiness of certain affine Deligne-Lusztig varieties associated to the general linear group. Formal groups, \(p\)-divisible groups, Classical groups (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Finite nilpotent groups, \(p\)-groups \(p\)-kernels occurring in an isogeny class of \(p\)-divisible 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 Fix \(p_1,\dots, p_h \in \mathbb{P}^r\) distinct points and fix \(m_1, \dots ,m_h\) positive integers. Let \({\mathcal{L}}_{r,d}\) be the linear system of hypersurfaces of \(\mathbb{P}^r\) of degree \(d\) and consider \[ {\mathcal{L}}:={\mathcal{L}}_{r,d}(m_1, \dots, m_d) \] the subsystem of those divisors of \({\mathcal{L}}_{r,d}\) having multiplicity at least \(m_i\) at \(p_i\), \(i=1, \dots, n\). Its virtual dimension is defined to be \[ \nu({\mathcal{L}}):={r+d \choose r}-1-\sum{i=1}^n {r+m_i-1 \choose r} \] i.e., the virtual dimension of \({\mathcal{L}}_{r,d}\) minus the number of conditions imposed by all multiple points \(p_i\). This number cannot be less than \(-1\), hence we define the expected dimension to be \[ \epsilon({\mathcal{L}}):=\text{max}\{\nu({\mathcal{L}}),-1\}. \] If the conditions imposed by the assigned points are not linearly independent, the effective dimension of \({\mathcal{L}}\) is greater than the expected one: in that case we say that \({\mathcal{L}}\) is special. Otherwise, if the effective and the expected dimension coincide, we say that \({\mathcal{L}}\) is non-special. What is known, as yet, is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in \(\mathbb{P}^r\) gives independent conditions on the linear system \({\mathcal{L}}\) of the hypersurfaces of degree \(d\), with a well known list of exceptions. In this paper the author presents a new proof of this theorem which consists in performing degenerations of \(\mathbb{P}^r\) and analyzing how \({\mathcal{L}}\) degenerates. The degenerations used here were introduced by \textit{C. Ciliberto} and \textit{R. Miranda} [J. Reine Angew. Math. 501, 191--220 (1998; Zbl 0943.14002)] and, originally proposed by Z. Ran, to study higher multiplicity interpolation problem. The original approach consists in degenerating the plane to a reducible surface, with two components intersecting along a line, and simultaneously degenerating the linear system \({\mathcal{L}}\) to a linear system \({\mathcal{L}}_0\) obtained as fibered product of linear systems on the two components over the restricted system on their intersection. The limit linear system \({\mathcal{L}}_0\) is somewhat easier than the original one, in particular this degeneration argument allows to use induction either on the degree or on the number of imposed multiple points. This contruction provides a recursive formula for the dimension of \({\mathcal{L}}_0\) involving the dimensions of the systems on the two components. In this paper the author generalizes this approach to the case with \(r \geq 3\) and completes the proof of Alexander-Hirschowitz Theorem with this method, exploiting induction on both \(d\) and \(r\). A tricky point of this approach is the study of the transversality of the restrictions of the systems on the intersection of the two components. In the planar case, Ciliberto and Miranda proved it using the finiteness of the set of inflection points of linear systems on \(\mathbb{P}^1\). In higher dimension transversality is more complicated. In Section 2.2 and in Section 3.1, the author presents a new approach to this problem: if at least one of the two restricted systems is a complete linear system, then the dimension of the intersection is easily computed. Anyhow, this is not sufficient to finish the proof of Alexander-Hirschowitz Theorem. For instance, it does not work in the cubic case. The solution to this obstacle is to blow up a codimension three subspace \(L\) of \(\mathbb{P}^r\), instead of a point. This approach to the cubic case is not so different from the one proposed by \textit{M. C. Brambilla} and \textit{G. Ottaviani} [J. Pure Appl. Algebra 212, No. 5, 1229--1251 (2008; Zbl 1139.14007)] where they give another alternative, and short, proof of Alexander-Hirschowitz Theorem, in the case \(d\geq 4\), and propose a new and simpler degeneration argument in the cubic case. Also the quartic case must be analysed separately. Indeed, twisting by a negative multiple of the exceptional component of the central fiber, one reduces to quadrics that are special. The proof, in this case, involves a geometric argument that exploits the property of cubics of containing all lines through two distinct double points. The constructions in this paper, besides its intrinsic intent, gives hope for further extensions to greater multiplicities. degenerations; polynomial interpolation; linear systems; double points Postinghel, E.; A new proof of the Alexander-Hirschowitz interpolation theorem; Ann. Mat. Pura Appl.: 2012; Volume 191 ,77-94. Divisors, linear systems, invertible sheaves, Fibrations, degenerations in algebraic geometry, Projective techniques in algebraic geometry A new proof of the Alexander-Hirschowitz interpolation theorem

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 \(M\) be the complement of a hypersurface \(V\) in the \(n\)-dimensional the projective space \(\mathbb{P}^n\). Let \(V_1,\dots,V_m\) be the irreducible components of \(V\). If \(L\) is a generic rank one local system on \(M\), it is well known that \(H^k(M,L)=0\) for all \(k \neq n\). In this paper the author gives some sufficient conditions on the components \(V_j\) and on the local system \(L\) such that two twisted cohomology groups are non-zero, namely \(H^k(M,L) \neq 0\) for \(k=n-1\) and \(k=n\). Explicit non-trivial cohomology classes are constructed using rational differential forms and some geometric examples are given. local systems; twisted cohomology; hypersurface arrangements de Rham cohomology and algebraic geometry, Divisors, linear systems, invertible sheaves, Relations with arrangements of hyperplanes Non-vanishing of the twisted cohomology on the complement of hypersurfaces

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 \(\delta\) be a locally nilpotent derivation on \(\mathbb C[x_1,\dots,x_n]\) such that the associated \(G_a\)-action is fixed point free. Suppose that the ring of invariants \(R := \mathbb C[x_1,\dots,x_n]^{\delta}\) is finitely generated. Let \(\mathfrak m \subset R\) be a maximal ideal such that \(R_{\mathfrak m}\) is a regular local ring and \(S := R \smallsetminus \mathfrak m\). It is proved that \(\Omega_{S^{-1}\mathbb C[x_1,\dots,x_n]| R_{\mathfrak m}}\) is free of rank one. A consequence is the following result: If \(R\) is regular then the associated \(G_a\)-action is locally trivial if and only if it is proper. Finally it is proved that a set of polynomials \(\{f_1,\dots,f_{n-1}\}\) is part of a coordinate system for \(\mathbb C^n\) (i.e. there exist \(f_n\) such that \(\mathbb C[x_1,\dots,x_n] = \mathbb C[f_1,\dots,f_n]\)) if and only if \(\mathbb C[f_1,\dots,f_{n-1}]\) is the ring of invariants for a proper \(G_a\)-action (the action is generated by the derivation \(\delta\) defined by \(\delta(h) = \lambda \det(J(f_1,\dots,f_{n-1},h))\) for some \(\lambda \in \mathbb C^*\)). additive group action; ring of invariants; nilpotent derivation James K. Deveney and David R. Finston, Regular \?\? invariants, Osaka J. Math. 39 (2002), no. 2, 275 -- 282. Actions of groups on commutative rings; invariant theory, Polynomial rings and ideals; rings of integer-valued polynomials, Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Regular \(G_a\) invariants.

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 motivation for this paper comes from study of connected simple normal crossing divisors (all whose irreducible components are smooth rational curves) on a smooth projective algebraic surface \(\mathbb{C}\). Assume further that the dual graph of the divisor is a linear chain. Using the notions of blowing up and blowing down (modelled from theory of surfaces), the author classifies all such linear chains. He shows that any such linear chain is equivalent to a canonical chain. The last expression means roughly a linear chain containing an initial segment where each irreducible component has self-intersection \(0\), followed by a linear chain where all self-intersection are \(\leq -2\). Because of Hodge index theorem not all such linear chains can occur on algebraic surfaces. Similarly, the author defines a prime class (special linear chains) and gives a classification of all liner chains in terms of these. weighted trees Daigle, D.: Classification of linear weighted graphs up to blowing-up and blowing-down, Canad. J. Math. 60, 64-87 (2008) Rational and ruled surfaces, Birational automorphisms, Cremona group and generalizations, Classification of affine varieties, Graph theory Classification of linear weighted graphs up to blowing-up and blowing-down

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{M. Baker} and \textit{S. Norine} [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)] introduced a theory of linear systems on graphs very similar to the theory of divisors on smooth complex varieties. \textit{M. Baker} [Algebra Number Theory 2, No. 6, 613--653 (2008; Zbl 1162.14018)] gave a specialization map from smooth complex curves to strongly semistable curves which explains the similarities. In the paper under review the author extends the theory (also for metrizable graphs and tropical curves) to the real case, i.e. the case of graphs with a real structure. This is not a pointless generalization, not only because it is useful, but also because among the possible definitions of real structure, this seems to be the right one. If \(X\) is a smooth projective curve defined over \(\mathbb {R}\), then it has a genus \(g(X)\) measured by the topological space \(X(\mathbb {C})\), the number \(s(X)\) of connected components of \(X(\mathbb {R})\) and an integer \(a(X)\) measuring if \(X(\mathbb {C})\setminus X(\mathbb {R})\) is connected or not. For a graph with a real structure \(G\) Coppens introduces a real locus \(G(\mathbb {R})\) and integers \(g(G)\), \(s(G)\) and \(a(G)\); the integer \(s(G)\) involves also the genera of the components of \(G(\mathbb {R})\). For real divisors on \(G\) he extends the main results (like parity invariant) of real linear systems on a real curve. He also look at some extremal case (\(X\) an \(M\)-curve) in which differences occur. real curve; real projective curve; stable curve; divisors on curves; divisors on graphs; real linear systems on graph; metrizable graph; tropical curve; \(M\)-curve Coppens, M.: Linear systems on graphs with a real structure. Q. J. Math. (to appear) Real algebraic and real-analytic geometry, Special divisors on curves (gonality, Brill-Noether theory), Special algebraic curves and curves of low genus, Graphs and linear algebra (matrices, eigenvalues, etc.), Paths and cycles Linear systems on graphs with a real structure

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 f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a holomorphic map with an isolated singularity at the origin, \(F=f^{-1}(z)\) with sufficiently small \(| z|\), \(\Lambda =H_ n(F,{\mathbb{Z}})\) the Milnor lattice. Replacing f by a linear perturbation \(\tilde f,\) one gets \(\mu\) ordinary double points, and if one connects the origin to the critical values of \(\tilde f,\) one gets a basis of \(\Lambda\) consisting of vanishing cycles. Any basis obtained from such a basis by a monodromy transformation is called a geometric basis [\textit{E. Brieskorn}: Proc. Symp. Pure Math. 40, Part 1, 153-165 (1983; Zbl 0527.14006)]. The main theorem of the paper under review states that if two singularities f and g have a common geometric basis then they have the same topological type. The proof consists of constructing an h-cobordism between the complements of the links of the singularities. This naturally forces the author to exclude the case \(n=2\). It is announced that this case shall be treated in another paper. Milnor fibre; isolated singularity; vanishing cycles S. Szczepanki, Geometric bases and topological equivalence , Comm. Pure Appl. Math. 40 (1987), 389-399. Local complex singularities, Singularities in algebraic geometry Geometric bases and topological equivalence

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 local normal ring of essentially finite type over a field of characteristic 0, with canonical module \(K_A\) and let \(f:X\to Y=\text{Spec} (A)\) be a resolution of singularities of \(A\). Assume that \(K_X=f^* (K_Y)+ \sum^s_{i=1} a_iE_i\), where \(E=\bigcup^s_{i=1} E_i\) is the exceptional divisor and \(a_i\in \mathbb{Q}\). From this relation follows the definition of different classes of singularities (terminal, canonical, log-terminal, log-canonical singularities). Analogous definitions are given via Frobenius morphism in characteristic \(p>0\) (F-terminal, F-canonical, F-regular, F-pure singularities). The author considers mod \(p\) reduction of a ring \(A\) essentially of finite type over a field of characteristic 0 and says that \(A\) has \(\text{F}^{**}\)-type if the reduction mod \(p\) of \(A\) has a singularity \(\text{F}^{**}\) for large \(p\). The aim of this paper is to study relations between \(\text{F}^{**}\)-type and \(\;^{**}\) singularities in characteristic 0. Several good examples are given. reduction modulo \(p\); tight closure; singularities; Frobenius morphism Kei-ichi Watanabe, Characterizations of singularities in characteristic 0 via Frobenius map, Commutative algebra, algebraic geometry, and computational methods (Hanoi, 1996) Springer, Singapore, 1999, pp. 155 -- 169. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry Characterizations of singularities in characteristic 0 via Frobenius map

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 field of fractions \(K\). Let \(\mathcal{A} \) be an Azumaya algebra over \(R\). From the paper: ``Our work was motivated by \dots{} a paper by \textit{I.A. Panin} and \textit{M. Ojanguren} [Math. Z. 237, No. 1, 181-198 (2000; Zbl 1042.11024)] on the Grothendieck conjecture for Hermitian spaces. The point was to offer a good axiomatization for the method used in that paper.'' The results of the paper are as follows: If \(R\) contains a field, then \(R^{\ast }/\)Nrd\((\mathcal{A}^{\ast })\rightarrow K^{\ast }/\)Nrd\((\mathcal{A}_{K}^{\ast })\) is injective; and the canonical map \(H_{\text{ét}}^{1}(R,\text{SL}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,\text{SL}_{1,\mathcal{A}_{K}})\) is injective. These results were originally proved by \textit{I. Panin} and \textit{A.A. Suslin} [St. Petersbg. Math. J. 9, No. 4, 851-858 (1998); translation from Algebra Anal. 9, No. 4, 215-223 (1997; Zbl 0902.16019)]. Here ``Nrd'' is the reduced norm. If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))\rightarrow U(C_{K})/ \text{Nrd}(U(\mathcal{A}_{K}))\) is injective and the canonical map \[ H_{\text{ét}}^{1}(R,\text{SU}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,SU_{1,\mathcal{A}_{K}}) \] is trivial. Here \(U(C)\) is the unitary group of the center of \(\mathcal{A}.\) For \(d\) a natural number, if \(R\) contains a field, then \(R^{\ast }/\)Nrd\(( \mathcal{A}^{\ast })(R^{\ast })^{d}\rightarrow K^{\ast }/\text{Nrd}(\mathcal{A} _{K}^{\ast })(K^{\ast })^{d}\) is injective. If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))U(C)^{d}\rightarrow U(C_{K})/\text{Nrd}(U(\mathcal{A} _{K}))U(C_{K})^{d}.\) This paper originally appeared in Russian. The English translation can be difficult to follow. regular local ring; Azumaya algebra; Grothendieck conjecture for Hermitian spaces Homogeneous spaces and generalizations, Regular local rings On Grothendieck's conjecture about principal homogeneous spaces for some classical algebraic 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 The starting point of the paper is the fact that controllable linear systems of dimension \(n\) over \(\mathbb C\) represent the stable points of the \(\text{GL}_n(\mathbb C)\)-action (i.e., the change of basis in the state space) with respect to a suitable linearization. The paper studies this kind of relationship for systems with one input and no output, represented as pairs \((v,\phi)\in V\times \text{End}(V)\), where \(V\) is an arbitrary \(n\) dimensional complex space and \(\phi \) is a trace-free endomorphism. The main result states that a system \((v,\phi)\) is stable iff \(v,\phi(v),\dots, \phi^{n-1}(v)\) are linear independent and \((v,\phi)\) is semistable iff it is stable or \(\phi^n\neq0\). An explicit description of the stable and semistable loci is provided in the case \(n=2\). stable locus; linear system; controllability Geometric methods, Controllability, Linear systems in control theory, Geometric invariant theory, Group actions on varieties or schemes (quotients) GIT-stability for a class of linear 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 article under review approaches the notion of systems given by a vector bundle on a Riemann surface (of finitely many connected components) together with a logarithmic connection and certain linear algebra data encoding the residues of the connection poles. These systems are called \(\lambda\)-connection systems. Theorem 2 shows that the category of \(\lambda\)-connection systems is equivalent, by the monodromy functor, to the category of representations of the punctured (by the poles of the connection) Riemann surface on the general linear group, with relations coming from those between the residues. For compact Riemann surfaces, the target of the monodromy functor is precisely the category of representations of a multiplicative preprojective algebra, constructed as representations of the path algebras of the double quiver. As a final application, it is shown that the multiplicative preprojective algebra of a Dynkin quiver is isomorphic to its usual preprojective algebra, and it can be proved a particular case of Hilbert's 21st problem. monodromy functor; logarithmic connections; multiplicative preprojective algebras; vector bundles on punctured Riemann surfaces; Hilbert's 21st problem Crawley-Boevey (W.).-- Monodromy for systems of vector bundles and multiplicative preprojective algebras. Bull. London Math. Soc. 45, p. 309-317 (2013). Vector bundles on curves and their moduli, Connections (general theory), Representations of quivers and partially ordered sets, Monodromy on manifolds Monodromy for systems of vector bundles and multiplicative preprojective 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 The author considers a flat algebraic group scheme G over a noetherian base scheme S acting on a noetherian scheme X. He proves that in certain cases, any sheaf of finitely generated modules with G-action on X admits an equivariant resolution by finitely generated locally free modules with G-action. The restrictions on X, S and G are of the following sort: X and S are regular or affine, or are quasiprojective over an affine scheme; G is semisimple, or is reductive over a normal base S, or G is affine and smooth with connected fibers over a regular noetherian base S of dimension \(\leq 2\). The statement about reductive groups is a conjecture of Seshadri, who showed that it implies finite generation of rings of invariants of reductive group actions [\textit{C. S. Seshadri}, Adv. Math. 26, 225-274 (1977; Zbl 0371.14009)]. The resolution results also yield proofs that semisimple groups over any base S (and certain other types of groups as well) are linear in that they can be embedded as closed subgroups of Aut(V) for V a vector bundle over S. Similarly, if X is non- equivariantly affine (resp. normal and quasiprojective over S) then X can be equivariantly embedded into a linear action on a vector space bundle (resp. projective space bundle), strengthening the results of \textit{H. Sumihiro} in J. Math. Kyoto Univ. 15, 573-605 (1975; Zbl 0331.14008). linearization of semisimple groups; flat algebraic group scheme; equivariant resolution Thomason, R. W., \textit{equivariant resolution, linearization, and hilbert's fourteenth problem over arbitrary base schemes}, Adv. Math., 65, 16-34, (1987) Group schemes, Group actions on varieties or schemes (quotients) Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base 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 The local models of \(k\)-analytic spaces in the sense of Berkovich are spectra of \(k\)-algebras of the form \[ k\langle r^{-1}_1X _1,\dots, r^{-1}_n X_n\rangle /I, \] where the radii of convergence \(r_v\) are arbitrary elements of \(\mathbb{R}^X_+\). If \(r_v\in \sqrt{|k^X|}\), such algebras are called strictly \(k\)-affinoid and the corresponding global spaces strictly \(k\)-analytic. The author proves that the functor of spaces \[ \text{strictly }k\text{-An}\to k\text{-An} \] is fully faithful, in particular there is at most one strict model of a \(k\)-analytic space. His main tool is a generalized concept of reduction of an affinoid space: \[ \widetilde A:= \bigoplus_{r\in \mathbb{R}^X_+} A_r/A_{<r}, \] where \(A_r\) resp. \(A_{<r}\) denotes the set of elements with spectral norm \(\leq r\) resp. \(<r\). The reduction \(\widetilde k\) is not a field, but as a graduated ring very much behaves like a field. Thus the author succeeds -- not without some technical difficulties -- in defining and studying thoroughly reductions of germs of analytic spaces and gets the desired result. A by-product is the statement that for \(k\)-analytic spaces the properties ``closedness'' and ``properness'' of morphisms are local with respect to the base space. For part I of this paper, see the author, Math. Ann. 318, No. 3, 585--607 (2000; Zbl 0972.32019). rigid analytic spaces; \(k\)-analytic spaces; graded valuation rings Michael Temkin, ``On local properties of non-Archimedean analytic spaces. II'', Isr. J. Math.140 (2004), p. 1-27 Non-Archimedean analysis, Rigid analytic geometry, General valuation theory for fields, Valuations and their generalizations for commutative rings On local properties of non-Archimedean analytic spaces. II

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 original work of Mather giving necessary and sufficient conditions for finite determinacy of smooth map-germs (e.g. up to right-left equivalence) gave very poor estimates of the degree of determinacy, although these have subsequently been considerably improved. The object of this paper is to give much sharper results with a good deal more flexibility. This allows, for example, applications to the equivariant case, to determinacy modulo a conveniently chosen subspace, and to bifurcation theory. Such estimates can be used as an effective tool in classification problems. Under certain circumstances, necessary and sufficient conditions are obtained. Although the method can be derived from an elementary iterative construction, it is most clearly expressed using properties of actions of unipotent algebraic groups. The known result here that orbits are closed leads (under appropriate conditions) to intimate relations between orbits and their tangent spaces which allows the derivation of the main results. Thus an affine subspace X of a vector space on which a unipotent group acts is contained in a single orbit provided the tangent space T(x) at some \(x\in X\) to its orbit contains all other T(y) as well as the tangent space to X. A typical result is that if G is a ''jet-closed, unipotent'' group of contact equivalences acting on \(C^{\infty}(n,p)\); M a vector subspace such that there is an induced action on the quotient, then f is M-G- determined if and only if, for some s, f is s-G-determined and \(J^ sM\subset LG.j^ sf\). finite determinacy of smooth map-germs; right-left equivalence; bifurcation theory; ''jet-closed, unipotent'' group Bruce J.W., du Plessis A.A., Wall C.T.C.: Determinacy and unipotency. Invent. Math. 88, 521--554 (1987) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Group actions on varieties or schemes (quotients), Group structures and generalizations on infinite-dimensional manifolds Determinacy and unipotency

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 perfect infinite field, and let \(\mathbb P^n(\overline k)\) be an \(n\)-dimensional projective space over the algebraic closure \(k\) of \(k\) with homogeneous coordinates \(X_0,\dots, X_n\). Consider an \((n-s)\)-dimensional algebraic variety \(V\subset\mathbb P^n(\overline k)\) defined and irreducible over \(k\), where \(1\leq s\leq n\), such that \(\deg V = D\). Let \(V\) be the set of all zeros of a homogeneous prime ideal \({\mathfrak P}\subset k[X_0,\dots,X_n]\). We prove that the local ring of \(V\) has a regular sequence (a sequence of local parameters if \(\text{char}(k) = 0\)) consisting of \(s\) nonhomogeneous polynomials such that the product of their degrees is less than \(D\) multiplied by some constant depending on \(n\). If \(V\) is a component of an algebraic variety given by a system of homogeneous polynomial equations of degree less than \(d\) over a field of characteristic 0, then the degrees of all these local parameters are less than \(d\) multiplied by some constant depending on \(n\). These constants depending on \(n\) can be estimated. A. L. Chistov, ''Efficient construction of local parameters of irreducible components of an algebraic variety,'' \textit{Proc. St.Petersburg Math. Soc.}, \textbf{7}, Amer. Math. Soc. Transl. Ser. 2, \textbf{203}, 201-231 (2001). Computational aspects of higher-dimensional varieties Effective construction of local parameters of irreducible components of an algebraic variety.

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 introduce the notion of a \textit{resolution supported on a poset}. When the poset is a CW-poset, i.e., the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by \textit{D. Bayer} and \textit{B. Sturmfels} [J. Reine Angew. Math. 502, 123--140 (1998; Zbl 0909.13011)]. Work of \textit{V. Reiner} and \textit{V. Welker} [Math. Scand. 89, No. 1, 117--132 (2001; Zbl 1092.13025)], and of \textit{M. Velasco} [J. Algebra 319, No. 1, 102--114 (2008; Zbl 1133.13015)], has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \textit{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and, generalizing results of Miller and Sturmfels, we prove a fundamental relationship between Artinianizations and Alexander duality for monomial ideals. Syzygies, resolutions, complexes and commutative rings, Algebraic aspects of posets, Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial aspects of commutative algebra Minimal free resolutions of monomial ideals and of toric rings are supported on posets

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 \(R/\mathfrak m\) an algebraically closed field of characteristic zero, and let \(J(\neq \mathfrak m)\) be a complete \(\mathfrak m\)-primary ideal in \(R\). The purpose of the paper is to study adjacent complete ideals above \(J\). (If a complete ideal \(I\) contains \(J\) and \(\text{length}(I/J)=1\), it is said that \(I\) is an adjacent complete ideal above \(J\).) The first main result (Theorem 3.12) is as follows: Every adjacent complete ideal above \(J\) is the integral closure of some ideal \((f,g)\) where \(f\) is a generic element of \(J\) and \(g\) has effective multiplicities previously determined in terms of the equisingularity class of \(J\). The second one (Theorem 4.2) gives a decomposition of the set \({\mathbf I}_J\) of adjacent complete ideals above \(J\) into a finite set of families, each one being associated to some Rees valuation of \(J\). As a corollary (Corollary 4.4), a condition for \({\mathbf I}_J\) to be finite is given. Finally (Subsections 4.1 and 4.2), the authors give an algorithmic procedure to generate the ideals in \({\mathbf I}_J\) in terms of their base points. Throughout the paper the use of the techniques of clusters of infinitely near points in dimension two as revised by \textit{E. Casas-Alvero} [Singularities of plane curves. London Mathematical Society Lecture Note Series. 276. Cambridge: Cambridge University Press (2000; Zbl 0967.14018)] plays a key role. two dimensional regular local ring; complete ideal; adjacent; cluster Other special types of modules and ideals in commutative rings, Regular local rings, Integral closure of commutative rings and ideals, Singularities in algebraic geometry On adjacent complete ideals above a given complete 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 vector space of dimension n over a field \(K\). Let \(N\) be the set of all nilpotent endomorphisms of \(V\) and let \(Y\) be the set of all flags \(F_*=(F_ i)_{0\leq i\leq n}\) in \(V\). Let \(N(F_*)=\{x\in N| xF_ i\subset F_{i-1}\text{ for all } i>0\}.\) Note that \(x\in N(F_*)\) if and only if x is strictly upper triangular with respect to a basis of V corresponding to \(F_*\). Let \(X=\{(x,F_*)\in N\times Y|\) \(x\in N(F_*)\}\) and let \(\pi: X\to N\) be the morphism given by the first projection. Let \(Y(x)\) denote the fiber \(\pi^{-1}(x)\). This paper considers a classification of the elements of \(X\), which can be thought of as strictly upper triangular matrices, and of the fiber \(Y(x)\), which is the conjugacy class of \(x\). A nilpotent endomorphism \(x\) is characterized by a partition \(\lambda (x)=(\lambda_ 1,...,\lambda_ r)\), where \(\lambda_ 1\geq...\geq \lambda_ r\) are the sizes of the Jordan blocks of \(x\). Given \((x,F_*)\), \(x\) induces nilpotent endomorphisms on the subquotients \(F_ q/F_ p\), hence a system \(t=(t[p,q])_{p<q}\) of partitions. The information contained in such a system of partitions is reorganized in the form of a strict upper triangular matrix with entries zeros or ones. Such a matrix is called a typrix. A typrix is said to occur if it arises from some \((x,F_*)\). The author obtains separate sets of conditions which are either necessary or sufficient for a typrix to occur. The sufficient conditions are used to describe certain dense subsets of the irreducible components of the fibers \(Y(x)\), in case K is infinite. The necessary conditions are of a combinatorial nature and are amenable to verification by a computer, if \(n\) is not too large. In this way the author obtains a list of possibly occurring typrices for \(n\leq 7\). For larger \(n\), a number of occurring typrices are described in several special cases. nilpotent endomorphisms; flags; classification; strictly upper triangular matrices; fiber; Jordan blocks; typrix Hesselink, W. H.: A classification of the nilpotent triangular matrices. Compositio math. 55, 89-133 (1985) Canonical forms, reductions, classification, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields A classification of the nilpotent triangular matrices

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 discusses four different constructions of vector space bases associated to vanishing ideals of points and he shows how to compute normal forms with respect to these bases giving new complexity bounds. Let \(k[x_1, \dots, x_n]\) be the polynomial ring in \(n\) variables over a field \(k\). The vanishing ideal \(I\) with respect to a set of points \(\{p_1, \dots, p_n\}\) in \(k^n\) is defined as the set of elements of \(k[x_1, \dots, x_n]\) that are zero on all of the \(p_i\)'s. The main tool that is used to compute vanishing ideals of points is the Buchberger-Möller algorithm which returns a Gröbner basis for the ideal vanishing on the set \(\{p_1, \dots, p_n\}\). A complementary result of the algorithm is a vector space basis for the quotient ring \(k[x_1, \dots, x_n]/I\). In several applications it turns out that the attention is more related on this vector space basis than in the Gröbner basis. For example, it may be preferable to compute normal forms using vector spaces methods instead of Gröbner basis techniques. Thus the author introduces four different approaches for the construction of the vector space basis, all of which perform better than the Buchberger-Möller algorithm. The first construction produces a vector space basis, for the quotient ring, given by a family of separators, that is, a family \(\{f_1, \dots, f_n\}\) of polynomials such that \(f_i(p_i)=1\) and \(f_i(p_j)=0\) if \(i\not= j\). The second construction is a \(k-\)basis formed by the residues \(1,f,\dots, f^{m-1}\), where \(f\) is a linear form. The third construction produces a set of monomials outside the initial ideal of \(I\) with respect to the lexicographical ordering, using only combinatorial methods. Finally, the fourth construction gives a \(k-\)basis which is the complement of the initial ideal with respect to a class of admissible monomial orders. A fundamental element for the effectiveness of these methods is a fast combinatorial algorithm which gives useful structure informations about the relations between the points. As an important application, the author drastically improves the computational algebra approach to the reverse engineering of gene regularity networks. ideals of points; normal form; standard monomials Lundqvist S.: Vector space bases associated to vanishing ideals of points. J. Pure Appl. Alg. 214(4), 309--321 (2010) Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Affine geometry Vector space bases associated to vanishing ideals 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 Let \(A\) be a regular connected semi-local ring containing a field. Denote by \(F\) the quotient field of \(A\) and assume that each residue field of \(A\) is infinite. The author's main theorem in the paper under review states that the canonical maps \(i_n: K^M_n(A)\to K^M_n(F)\) of the corresponding Milnor \(K\)-groups are universally injective, for all integers \(n\geq 0\) in this particular situation. The proof of this major theorem is based on several original new ingredients, including a Néron-Popescu desingularization method, the construction of a certain co-Cartesian square of Milnor \(K\)-groups motivated by motivic cohomology, and a generalization of the Milnor-Bass-Tate sequence in algebraic \(K\)-theory to semi-local rings. As it turns out, the author's main theorem has various important consequences. A first application leads to the conclusion that the so-called Gersten conjecture is true in an equivariant context. More precisely, the author derives a theorem stating that \textit{K. Kato}'s Gersten complex of Zariski sheaves for Milnor \(K\)-theory of an excellent scheme \(X\) is exact if \(X\) is regular over an infinite field. In the sequel, the exactness of the Gersten complex is used to prove one of the still remaining Beilinson conjectures on motivic cohomology. In fact, it is shown that for Voevodsky's motivic complexes of Zariski sheaves \(\mathbb{Z}(n)\) on the category of smooth schemes over an infinite field, there is an isomorphism \({\mathcal K}^M_n\overset\sim\rightarrow{\mathcal H}^n(\mathbb{Z}(n))\) for all \(n\geq 0\). As a further consequence of the verified Gersten conjecture, the author deduces a Bloch formula relating Milnor \(K\)-theory and Chow groups of (excellent) schemes \(X\) in the form \(H^n(X,{\mathcal K}^M_n)\simeq CH^n(X)\) for all \(n\geq 0\). Finally, it is pointed out how \textit{M. Levine}'s generalized Bloch-Kato conjecture [``Relative Milnor \(K\)-theory'', K-Theory 6, No.~2, 113--175 (1992; Zbl 0780.19005)] can be turned into a true theorem for semi-local rings containing an infinite field of suitable characteristic, as well as how the generalized Milnor conjecture on quadratic forms over local rings can be verified. In an appendix to the present paper, the author provides a generalization of a factorization result used by \textit{O. Gabber} in his earlier proof [Manuscr. Math. 95, No.~1, 107--115 (1998; Zbl 0896.13004)] of the surjectivity of the homomorphism of sheaves \({\mathcal K}^M_n\to{\mathcal H}^n(\mathbb{Z}(n))\) on the big Zariski site of smooth varieties over an infinite field. No doubt, the work under review must be seen as a highly important contribution to algebraic \(K\)-theory and its applications to algebraic geometry, as the author solves a number of long-standing problems simultaneously by a very original and rather unified approach, with his main theorem (Theorem 6.1.) being the decisive clue. Milnor \(K\)-theory; Gersten conjecture; motivic cohomology; Beilinson's conjecture; Chow groups; Bloch-Kato conjecture Kerz, Moritz, The Gersten conjecture for Milnor \(K\)-theory, Invent. Math., 175, 1, 1-33, (2009) Higher symbols, Milnor \(K\)-theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Motivic cohomology; motivic homotopy theory, (Equivariant) Chow groups and rings; motives, Étale and other Grothendieck topologies and (co)homologies The Gersten conjecture for Milnor \(K\)-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 A be the homogeneous coordinate ring of a union of s (straight) lines in the projective space \({\mathbb{P}}^ n_ k\), k a field. If \({}^+A\) is the normalization of A and if \(A_ i\) (resp. \({}^+A_ i)\) denotes the degree part of A (resp. \({}^+A)\), it was proved, by \textit{B. H. Dayton} and the author [Algebraic K-theory, Proc. Conf., Evanston 1980, Lect. Notes Math. 854, 93-123 (1981; Zbl 0464.14010)] that if the inclusion \(A_ i\to^+A_ i\) is an isomorphism for all \(i\leq s-1\), then A is seminormal. In another paper, by the same two authors, it was proved that \(\dim_ k(^+A/A)<\infty\) if and only if the directions of the lines are linearly independent at each intersection point. The purpose of the paper under review is to tie these two results as in the following theorem: Let A be the homogeneous coordinate ring of a union X of s straight lines in \({\mathbb{P}}^ n_ k\), such that the directions of the lines at each intersection point, are linearly independent, and let \({}^+A\) be the seminormalization of A. Then the inclusion \(f_ i: A_ i\to^+A_ i\) is an isomorphism for \(i\geq s-1\). If X is connected, then \(f_{s-2}\) is also an isomorphism. - Moreover, with examples, it is shown that the bounds s-1 and s-2 in the above theorem, in general, are the best possible and with other examples, it is shown that, in particular cases, such bounds can be improved. An attempt to explain how the result of the above theorem can be improved, in order to take in account all the above examples, is also given. union of straight lines; homogeneous coordinate ring; normalization Roberts, L. G.: Seminormalization of a union of lines. CR math. Rep. acad. Sci. Canada 9, 37-41 (1987) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, General commutative ring theory, Relevant commutative algebra, , Special algebraic curves and curves of low genus The seminormalization of a 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 Let \(\Delta\) be an irreducible root system on an Euclidean space \(E\) over \(\mathbb{R}\) and let \(\mathbb{P}(E_\mathbb{C})\) be the complex projective space associated to \(E\). For each subroot system of type \(A_3\) in \(\Delta\), it is possible to define an \(A_3\)-cross ratio map of \(Z(\Delta)\) to \(CR(\mathbb{P})\), where \(Z(\Delta)\) is a Zariski open subset of \(\mathbb{P}(E_\mathbb{C})\) and \(CR(\mathbb{P}) \simeq \mathbb{P}^1\) [for the precise definition of \(Z(\Delta)\) and \(CR(\mathbb{P})\) see \textit{J. Sekiguchi}, Kyushu J. Math. 48, No. 1, 123-168 (1994; Zbl 0841.14009)]. By taking the product of the \(A_3\)-cross ratio maps for all subroot systems of type \(A_3\) in \(\Delta\), we obtain a map \(cr_{\Delta, A_3}\) of \(Z(\Delta)\) to \(CR (\mathbb{P})^m\), where \(m\) is the number of subroot system of type \(A_3\) in \(\Delta\). We put \({\mathcal C}' (\Delta, A_3)= cr_{\Delta, A_3} (Z(\Delta))\) and denote by \({\mathcal C} (\Delta, A_3)\) its Zariski closure in \(CR(\mathbb{P})^m\). We now assume that \(\Delta =\Delta (A_{n+2})\) is of type \(A_{n+2}\). In this case, it is easy to see that \(\dim ({\mathcal C} (\Delta,A_3))=n\) and that \({\mathcal C} (\Delta,A_3)\) is regarded as a compactification of the complement of the hypersurface \({\mathcal S}_n\) in \(\mathbb{C}^n\) defined by \(\prod^n_{j=1} \{z_j (1-z_j)\} \prod_{i<j} (z_i-z_j)=0\), where \(z=(z_1, \dots, z_n)\) is a standard affine coordinates system of \(\mathbb{C}^n\). On the other hand, there is a natural compactification of \(\mathbb{C}^n \setminus {\mathcal S}_n\) constructed by \textit{T. Terada} [J. Math. Soc. Japan 35, 451-475 (1983; Zbl 0506.33001)] which is called the \((n\)-dimensional) Terada model and denoted \({\mathcal T}_n\) in this article. Moreover, both \({\mathcal C} (\Delta,A_3)\) and \({\mathcal T}_n\) admit \(W(A_{n+2})\)-actions. Noting these, we are led to ask whether \({\mathcal C} (\Delta,A_3)\) is isomorphic to \({\mathcal T}_n\) or not (cf. conjecture 2.2 (i) in the author's paper cited above). The purpose of this article is to give an answer affirmative to the question above, namely, to prove that \({\mathcal C} (\Delta,A_3)\) is isomorphic to \({\mathcal T}_n\) for each \(n\). cross ratio; root system; Terada model Sekiguchi, J, Cross ratio varieties for root systems of type \(A\) and the Terada model, J. Math. Sci. Univ. Tokyo, 3, 181-197, (1996) \(n\)-folds (\(n>4\)), Algebraic moduli problems, moduli of vector bundles, Rational and birational maps, Projective techniques in algebraic geometry, Hypersurfaces and algebraic geometry Cross ratio varieties for root systems of type \(A\) and the Terada model

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 \(A\) denote a local ring that admits a surjection from an \(m\)-dimensional regular local ring \((R,\mathfrak{m})\) containing its residue field \(k\). Let \(I\) denote the kernel of the surjection. For given nonnegative integers \(i,j\) the Lybeznik number \(\lambda_{i,j}(A)\) is defined as the \(i\)-th Bass number of \(H^{m-j}_I(R)\) with respect to \(\mathfrak{m}\). They were introduced by \textit{G. Lyubeznik} [Invent. Math. 113, No. 1, 41--55 (1993; Zbl 0795.13004)]. In the present paper the author completely determines \(\lambda_{i,j}(A)\) of a local ring \(A\) at the vertex of the affine cone over a nonsingular projective variety \(V\) over a field of characteristic zero. This is done in terms of the dimensions of the algebraic de Rham cohomology spaces of \(V\). It follows that these numbers are intrinsic numerical invariants of \(V\) even though a priori their definition depends on an embedding into projective space. Lyubeznik number; de Rham cohomology Switala, Nicholas, Lyubeznik numbers for nonsingular projective varieties, Bull. Lond. Math. Soc., 47, 1, 1-6, (2015) Local cohomology and commutative rings, de Rham cohomology and algebraic geometry Lyubeznik numbers for nonsingular projective 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 This paper answers affirmatively the question in the survey article by \textit{P. A. Griffiths} [``An introduction to the theory of special divisors on algebraic curves'', Regional Conf. Ser. Math. 44 (1980; Zbl 0446.14010)]: Does the variety \(W_d^r\) of linear systems on a general curve of genus \(g\) with degree \(d\) and dimension at least \(r\) have the Brill-Noether dimension \(g - (r +1)(g - d +r)\)? Moreover the authors determine the class of this variety in the cohomology of the Jacobian and show that it is without multiple components. The proof is by a detailed geometrical analysis of a classical degeneration idea of Castelnuovo's. It is formalized as the Castelnuovo-Severi-Kleiman conjecture (CSK): the family of \(P^k\)'s in \(P^d\) meeting the chords of a rational normal curve in \(P^d\) has the same dimension as if the chords were lines in general position, and the family has no multiple components. The paper has three parts: (I) The reduction to CSK -- except for the absence of multiple components to \(W_d^r\); (II) The proof of CSK; (III) Proofs of the absence of multiple components. (I) has been previously achieved by \textit{S. Kleiman} [Adv. Math. 22, 1--31 (1976; Zbl 0342.14012)]. The proof in this paper is by a geometrical argument based on duality of special divisors. (II) uses a degeneration of the chords to span an osculating flag. (III) is again by degeneration. The degenerate case is chosen so that number of intersections of two varieties as a set equals the algebraic intersection number. The techniques are those of classical algebraic geometry and Schubert calculus. variety of linear systems on a general curve; cohomology of Jacobian; Castelnuovo-Severi-Kleiman conjecture; algebraic intersection number; Schubert calculus Griffiths, P. \&amp; Harris, J.,On the variety of special linear systems on a general algebraic curves, Duke Math. J.,47(1980), 233--272. Jacobians, Prym varieties, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of special linear systems on a general algebraic 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 In its familiar form, the Kodaira vanishing theorem is a statement about the cohomology of the sheaf of sections of a positively curved line bundle \({\mathcal L}\) on a complex projective manifold \(X\); it asserts that \(H^i(X, {\mathcal L}^{-1}) =0\) for \(i\) less than the dimension of \(X\). However, the Kodaira vanishing theorem also has a purely ideal-theoretic interpretation concerning primary decompositions of ideals generated by part of a system of parameters for the section ring determined by \({\mathcal L}\). Moreover, the Kodaira vanishing theorem has a tight closure formulation, tying this classic and central result in complex algebraic geometry to new advances in characteristic \(p\) commutative algebra. In this paper, these algebraic formulations of the Kodaira vanishing theorem are developed. Our tight closure reinterpretation of Kodaira vanishing leads to a natural generalization, not at all apparent from any other point of view. The phrase ``\(S\) is a graded ring'' will mean that \(S=\bigoplus_{i\in\mathbb{N}}S_i\) is a finitely generated \(\mathbb{N}\)-graded algebra over the subring \(S_0\), which is a field. The irrelevant ideal \(\bigoplus_{i>0}S_i\) will be denoted by \(m\); we will use the notation \((S,m)\) to recall this convention. For any graded \(S\)-module \(M\), the notation \([M]_{\geq n}\) indicates the submodule of \(M\) consisting of those elements of degree \(n\) or more. A system of parameters is a set of \((\text{dim} S)\) elements generating an ideal primary to \(m\), whereas a set of \(i\) elements is called a set of parameters if it generates an ideal of height \(i\). All parameters and systems of parameters in this paper will be assumed to consist only of homogeneous elements. A parameter ideal for \(S\) is any ideal generated by parameters. Fix any set of parameters \(x_1, \dots, x_l\) for \(S\). Let \(I\) denote the ideal they generate and let \(x\) denote their product. The local cohomology module \(H_I^i(S)\) can be computed as the cohomology at the \(i\)th spot of the graded Čech complex \[ 0\to S\to \bigoplus S_{x_i} \to \bigoplus S_{x_{i_1} x_{i_2}} \to\cdots \to S_{x_1x_2 \dots x_l} \to 0. \] The notation \(\eta= \left[{z\over x^t} \right]\) will be used to denote an arbitrary element of \(H^l_I(S)\) (that is, an equivalence class in the cokernel of the last non-zero map in the Čech complex above). The notation \((I)^{ \text{lim}}\) indicates the ideal of all elements \(z\in S\) for which there exists an integer \(s\) with \((x)^{s-1} z\in(x^s_1, x^s_2, \dots,x^s_l)\). As long as the chosen generators for \(I\) are minimal, this does not depend on the choice of generators. Note that if the elements \(x_1,\dots,x_l\) form a regular sequence then \((I)^{\text{lim}} =IS\). The point of this closure operation is that \[ z\in (x_1,x_2, \dots, x_l)^{ \text{lim}} \] if and only if the element \(\left[ {z\over (x_1x_2 \dots x_l)} \right] \in H^l_I(S)\) represents the zero cycle. Definition. Let \(I\) be an ideal in a Noetherian ring \(R\) of characteristic \(p>0\). An element \(z\in R\) is in the tight closure of \(I\) if there exists an element \(c\in R\) (but not in any minimal prime of \(R)\) such that the following holds: for all \(q=p^e \gg 0\), the element \(cz^q\) is in the ideal generated by the \(q\)-th powers of all elements of \(I\). The set of all elements in the tight closure of \(I\) is denoted \(I^*\); it is easily seen that \(I^*\) is an ideal containing \(I\). - Tight closure for rings containing a field of characteristic zero is defined by reduction to characteristic \(p\). Vanishing conjecture. Let \(S\) be an equidimensional graded ring over a field \(S_0\) of characteristic 0. Assume that \(S\) has rational singularities away from the irrelevant ideal \(m\). Let \(x_1,\dots,x_d\) be any homogeneous system of parameters for \(S\). Then \[ (x_1,x_2, \dots, x_d)^*= (x_1, x_2, \dots, x_d)^{\text{lim}}+ S_{\geq\delta} \] where \(\delta\) is the sum of the degrees of the \(x_j\)'s. Theorem (Equivalent forms of the vanishing conjecture). The following are equivalent for an \(\mathbb{N}\)-graded Noetherian domain \((S,m)\) over a field \(S_0\) of characteristic \(p>0\) and of dimension \(d\). (1) \((x_1,x_2, \dots, x_d)^*\subset (x_1,x_2, \dots, x_d)^{\text{lim}}+ S_{\geq\delta}\) for all homogeneous systems of parameters \(x_1, \dots, x_d\) for \(S\) where \(\delta\) is the sum of the degrees of the \(x_j\)'s. (2) \((x_1, x_2, \dots, x_d)^*= (x_1,x_2, \dots, x_d)^{\text{lim}} +S_{\geq \delta}\) for all homogeneous systems of parameters \(x_1, \dots, x_d\) for \(S\) where \(\delta\) is the sum of the degrees of the \(x_j\)'s. (3) The tight closure of zero in \(H^d_m(S)\) has no nonzero elements of negative degrees. In particular, the tight closure of zero in \(H_m^d(S)\) is precisely the submodule of elements of nonnegative degrees. (4) The annihilator of the tight closure of zero in \(H^d_m(S)\) is \(m\)-primary and Frobenius acts injectively on \(H^d_m(S)\) in negative degrees. We call these generalizations of Kodaira's vanishing theorem simply the ``vanishing conjecture''. The vanishing conjecture is shown to imply the Kodaira vanishing theorem, to prove a long standing open problem in the theory of tight closure, and to give a sharp new Briançon-Skoda theorem for Gorenstein singularities. We also prove our vanishing conjecture, this ``strong'' form of Kodaira vanishing, holds in dimension two. Our interpretation of Kodaira vanishing in terms of tight closure grew out of a desire to better understand the relationship between analytic and characteristic \(p\) methods. Though the Kodaira vanishing theorem was first proved using the machinery of complex analytic differential geometry later proofs using reduction to characteristic \(p\) were discovered. Other theorems that have been proved by both analytic and characteristic \(p\) techniques include the new intersection theorem, the syzygy theorem, that rings of invariants have rational singularities, and the Briançon-Skoda theorem. More recently, the proof of Fujita's freeness conjecture for globally generated ample line bundles used tight closure techniques to establish in arbitrary characteristic what could previously be done only in characteristic zero with vanishing theorems. There is an interesting and as yet mysterious connection between the Frobenius operator in characteristic \(p\) and differential techniques over \(\mathbb{C}\); we hope this work sheds some additional light on this topic. characteristic \(p\); Kodaira vanishing theorem; tight closure; system of parameters; local cohomology Huneke, C.; Smith, K. E.: Tight closure and the Kodaira vanishing theorem. J. reine angew. Math. 484, 127-152 (1997) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Vanishing theorems in algebraic geometry, Integral closure of commutative rings and ideals, Local cohomology and commutative rings Tight closure and the Kodaira vanishing theorem

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 Katz has constructed some rigid local systems of rank 7 on \(\mathbb G_m\) in finite characteristics, whose monodromy is a finite subgroup \(G\) of a complex Lie group \(M\) of type \(G_2\). Among the five subgroups \(G\) of \(M\) which appear in his work, four occur in their natural characteristic: \[ \mathrm{SL}_2(8) \text{ in }\operatorname{char} 2,\quad \mathrm{PU}_3(3) \text{ in } \operatorname{char} 3, \quad \mathrm{PGL}_2(7) \text{ in }\operatorname{char} 7, \quad \mathrm{PSL}_2(13) \text{ in }\operatorname{char} 13. \] In this paper, using the theory of Deligne-Lusztig curves, the author gives another construction of these four local systems on \(\mathbb G_m\), and finds similar local systems with finite monodromy in other complex Lie groups. Some examples of finite groups which occur in exceptional complex Lie groups \(M\), analogous to the ones above are: \[ \begin{alignedat}{2}2 G &= \mathrm{PSL}_2(27) \text{ in }\operatorname{char}3,\quad& M&=F_4,\\ G &= \mathrm{PSU}_3(8) \text{ in }\operatorname{char}2,\quad& M&=E_7,\\ G &=\mathrm{PGL}_2(31) \text{ in }\operatorname{char}31,\quad& M&=E_8,\\ G &= \mathrm{PSL}_2(61) \text{ in }\operatorname{char}61,\quad& M&=E_8. \end{alignedat} \] There are interesting families in the classical groups. For example, when \(q=p^f>2\), we have \[ G=\mathrm{PU}_3(q) \text{ in }\operatorname{char}p,\quad M=\mathrm{Sp}_{2n},\quad 2n=q(q-1). \] As a bonus, the author obtains simple wild parameters for the local field \(k((1/t))\), by restricting the local systems to the decomposition group at \(t=\infty\). These parameters are representations of the local Galois group into \(M\), which have no inertial invariants on the adjoint representation and have Swan conductor equal to the rank of \(M\). rigid local systems; Deligne-Lusztig curves; finite simple subgroups of complex Lie groups; wild parameters; rigid embeddings; wild inertia group; Swan conductors; Galois group Gross, B. H., Rigid local systems on G_{\textit{m}} with finite monodromy, Adv. Math., 224, 6, 2531-2543, (2010) Linear algebraic groups over finite fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Rigid local systems on \(\mathbb G_m\) with finite monodromy

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 a series of two papers [for part II, cf. J. Pure Appl. Algebra 214, No. 5, 548--564 (2010; Zbl 1189.14034)] the authors describe the closure of the orbit of a plane curve under the action of the projective linear group. The knowledge of the boundary is useful for computing several characteristic numbers of families of plane curves. The target of the paper under review (the first one in the sequel) is a set-theoretical description of limit curves, arising when the matrix of the linear map becomes singular. Let \(f\) be the natural rational map from the space of matrices \(\mathbb P^8\) to the space \(\mathbb P^D\), parametrizing plane curves of degree \(d\), which define one orbit. The authors' main result is a description of the exceptional divisor, obtained by resolving the singularities of \(f\). The description depends on five features of limit curves \(C\) and their supports \(C'\), namely: the linear components, the non-linear components, the points at which the tangent cone of \(C\) is supported on at least three lines, the Newton polygon of \(C\) at singular and inflection points of \(C'\) and the Puiseux expansion of branches of \(C\) at singular points of \(C'\). plane curves Aluffi, P.; Faber, C.: Limits of \(PGL(3)\)-translates of plane curves, I, J. pure appl. Algebra 214, No. 5, 526-547 (2010) Plane and space curves, Rational and birational maps Limits of PGL(3)-translates of plane curves. 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 \(R\) be a semilocal commutative ring, and suppose throughout that \(2\) is invertible in \(R\). If \(A\) is an Azumaya algebra over \(R\) and \(\sigma\) is an \(R\)-linear involution of \(A\), then it is possible to define both \(\mathrm{O}(A, \sigma)\) -- the group of elements \(a \in A\) such that \(a^\sigma a = 1\) -- and \(\mathrm{SO}(A, \sigma)\), which is the kernel of the reduced norm map from \(O(A, \sigma)\) to the group \(\mu_2(R)\) of square-roots of \(1\) in \(R\). It should be noted that \(\mu_2(R) = \{-1, 1\}\) if \(R\) is connected. This short paper establishes two properties of \(\mathrm{O}(A, \sigma)\) and \(\mathrm{SO}(A, \sigma)\). Theorem 1 is that \(\mathrm{O}(A, \sigma)\) contains elements that have reduced norm \(-1\) if and only if the Brauer class of \(A\) is trivial. Theorem 2, which is used in proving Theorem 1, is that the natural map \(\mathrm{SO}(A, \sigma) \to \prod_{\mathfrak m} SO(A/\mathfrak m A, \sigma)\) is surjective, where \(\mathfrak m\) runs over the maximal ideals of \(R\). The first main theorem generalizes a result that is already known when \(R\) is a field, and the second generalizes a result of \textit{M. Knebusch}'s Satz 0.4, Lemma 3.1 of [Math. Z. 108, 255--268 (1969; Zbl 0188.35502)] in which \(A\) is assumed to be a matrix algebra. The proofs of the theorems rely on elements called \textit{reflections} in unitary groups (of which the orthogonal groups are a special case). Reflections were studied in much greater generality by the same author in [J. Pure Appl. Algebra 219, No. 12, 5673--5696 (2015; Zbl 1356.11019)] (generalizing constructions of [\textit{H. Reiter}, J. Algebra 35, 483--499 (1975; Zbl 0306.16017)]), and this paper is in part a pleasant working-out of that theory in a special case. The paper also contains some examples to show that the ``if'' direction of Theorem 1 fails in general if \(R\) is not assumed to be semilocal, and that Theorem 2 fails for the orthogonal, as distinct from the special-orthogonal, group. This contrasts with the case proved by Knebusch. The author conjectures that if \(R\) is a commutative ring (no longer assumed semilocal) such that \(2\) is invertible in \(R\), then \(O(A,\sigma)\) can contain elements of reduced norm \(-1\) only if \(A\) has trivial Brauer class. central simple algebra; Brauer group; involution; orthogonal group; Azumaya algebra; reduced norm Classical groups, Bilinear and Hermitian forms, Group schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) On the non-neutral component of outer forms of the orthogonal 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 This note is devoted to give a characterization of d-dimensional linear systems of quadrics in \({\mathbb{P}}^ r\), with Jacobian matrix of rank r- \(k\leq d\) and \(d\leq r\). The author divides such systems in two types: the irreducible and the reducible ones. For the first ones the characterization is as follows: the quadrics of the system passing through a general point of \({\mathbb{P}}^ r\) have a \({\mathbb{P}}^{k+1}\) in common. An analogous property holds for reducible systems. Though it is never stated explicitly, the base field has to be assumed of characteristic zero. It seems to the reviewer the exposition is unfortunately rather obscure. \(d\)-dimensional linear systems of quadrics L. Degoli: Trois nouveaux théorèmes sur les systèmes linéaires de quadriques à Jacobienne identiquement nulle. Demonstratio Mathematica, Warszawa, Vol. 16) Projective techniques in algebraic geometry, Projective analytic geometry, Divisors, linear systems, invertible sheaves Trois nouveaux théorèmes sur les systèmes linéaires de quadriques à Jacobienne identiquement nulle