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It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This is a survey of some results on the structure and classification of normal analytic compactifications of \(\mathbb C^2\). Mirroring the existing literature, we especially emphasize the compactifications for which the curve at infinity is irreducible. normal analytic surfaces; compactifications of \(\mathbb C^2\) Normal analytic spaces, Compactifications; symmetric and spherical varieties, Rational and ruled surfaces Normal analytic compactifications of \(\mathbb C^2\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This book considers the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution and studies some topics in representation theory from this more general context. The first goal of the authors is to apply these considerations to quiver varieties and hyperbolic varieties, and also to other quantized symplectic resolutions. The first part is devoted to the theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology. In a second part, the authors define and study the category O for a symplectic resolution and observe that this category is often Koszul. Then they define the notion of a symplectic duality between symplectic resolutions. This definition leads to a conjectural identification of two geometric realizations due to Nakajima and Ginsburg-Mirkovic-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups. An appendix by Ivan Losev provides a key step for the proof that the category O is of highest weight. quantizations; quantized symplectic resolutions; conical symplectic resolutions; Lie algebras; Harish-Chandra bimodules; cohomology; Koszul duality Braden, T.; Licata, A.; Proudfoot, N.; Webster, B., Quantizations of conical symplectic resolutions II: category O and symplectic duality, Astérisque, 384, 75-179, (2016) Research exposition (monographs, survey articles) pertaining to differential geometry, Deformation quantization, star products, Global theory and resolution of singularities (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Poisson algebras, Representation theory of associative rings and algebras Quantizations of conical symplectic resolutions

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties It was shown by \textit{C. O. Kiselman} [Acta Math. 117, 1-35 (1967; Zbl 0152.07602)] that the complement \({\mathbb P}^n \setminus V\) to an algebraic hypersurface \(V\) in a complex projective space \({\mathbb P}^n\) is a Stein manifold. In the article under consideration, the question is discussed about the Stein property of the complement \(M^n \setminus V\) to an algebraic surface \(V\) in an arbitrary toric variety \(M^n\). It is shown that, given a polynomial function \(F(z)\) of affine coordinates \(z \in {\mathbb C}^n\), it is possible to construct a canonical (native) toric variety \(M^n\), a compactification of the space \({\mathbb C}^n\), for which \(M^n \setminus V\) is a Stein manifold, where \(V\) is the closure of the hypersurface \(\{ z \in \mathbb C^n : F(z) = 0 \}\) in \(M^n\). This result is a generalization of the corresponding result of the author for the case of curves \(V\) in \({\overline{\mathbb C}}^2\). Stein property; toric variety; Newton polyhedron Yakovleva, O. V., On the Stein property of the complement of an algebraic surface in a toric manifold, Sibirsk. Mat. Zh., 39, 3, 703-713, (1998) Stein spaces, Toric varieties, Newton polyhedra, Okounkov bodies, Compactification of analytic spaces On the Stein property of the complement to an algebraic hypersurface in a toric variety

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties One can regard the projective space \({\mathbb{P}}^n\) as the variety representing the functor taking \(Y\) to surjections \({\mathcal O}_Y^{n+1}\to{\mathcal L}\), \({\mathcal L}\) a line bundle on \(Y\), since a map \(Y\to{\mathbb{P}}^n\) is given by \(n+1\) sections of a suitable \({\mathcal L}\) that do not all vanish simultaneously. In view of the very close analogy between toric varieties over a field~\(k\) and projective spaces, established by \textit{D. A. Cox} [J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)], it is reasonable to expect a good description of morphisms into toric varieties more generally. Several results of this nature are indeed known. Hitherto the most general has been one due also to \textit{D. A. Cox} [Tôhoku Math. J., II. Ser. 47, No. 2, 251-262 (1995; Zbl 0828.14035)], in which the toric variety \(X_\Delta\) is assumed to be smooth. Similar theorems, giving more directly applicable results but with strong conditions on~\(Y\) as well as~\(X_\Delta\), have been proved by \textit{M. A. Guest} [Bull. Am. Math. Soc., New Ser. 31, No. 2, 191-196 (1994; Zbl 0832.55009)] and by Tadao Oda and the reviewer. [This last result, unpublished at the time the present paper was written, has since appeared as \S 2 of the following: \textit{G. K. Sankaran}, Math. Ann. 313, No. 3, 409-427 (1999; Zbl 0919.14009){}. The author of the paper under review has pointed out an error in \S 1 of that paper, but \S 2 is unaffected.] In this context one should also mention \textit{M. Audin}, in whose book [``The topology of torus actions on symplectic manifolds'' (Birkhäuser 1991; Zbl 0726.57029)] on the topology of torus actions some of these results are foreshadowed. All these results are unified and generalised in the present paper. There is no hope of obtaining a good description on morphisms \(Y\to X_\Delta\) by means of invertible sheaves in all cases, because there are pathological toric varieties when for instance \(\text{Pic } X_\Delta\) is trivial. Hence the need to impose the condition ``enough effective Cartier divisors'', which means that the complement of every \(T\)-affine open subset is the support of an effective Cartier divisor. This condition is quite weak: All quasi-projective or simplicial toric varieties fulfil it. Given such an \(X_\Delta\), the author first studies the homogeneous coordinate ring \(S_\Delta\), which is the monoid ring on the monoid of effective Cartier divisors. He shows, following Cox's paper in J. Algebr. Geom. [loc.cit.], that \(X_\Delta\) is naturally isomorphic to a geometric quotient of an open subscheme of \(\mathop{\text{Spec}} S_\Delta\). The ring \(S_\Delta\) is \(\Delta\)-graded (i.e. has a \(\mathop{\text{Pic}}X_\Delta\)-grading) but, as the author points out, it is not possible to identify \(X_\Delta\) with the homogeneous spectrum of \(S_\Delta \) in general because \(\mathop{\text{Pic}} X_\Delta\) may have torsion. The author gives a necessary and sufficient condition for this torsion to vanish, incidentally correcting a minor error in Fulton's book on toric varieties [\textit{W. Fulton}, ``Introduction to toric varieties'' (Princeton 1993; Zbl 0813.14039)]. The next step is to establish a correspondence between \(\Delta\)-graded \(S_\Delta\)-modules and quasi-coherent \({\mathcal O}_\Delta\)-modules, largely a matter of verifying that proofs of similar results for smooth varieties still work. This is enough to prove the main general result: If \(Z\subset X_\Delta\) is a closed subscheme given by a homogeneous ideal \(I\) in \(S_\Delta\), and \(Y\) is a \(k\)-scheme, then \(k\)-morphisms \(Y\to Z\) correspond to \(\Delta\)-graded \({\mathcal O}_Y\)-algebra homomorphisms \(\phi:{\mathcal O}_Y\otimes_k S/I\to {\mathcal L}\) satisfying a nondegeneracy condition at each point of~\(Y\). Here \({\mathcal L}\) is a \(\Delta\)-graded \({\mathcal O}_Y\)-algebra with invertible components, so the product map \({\mathcal L}_\alpha\otimes{\mathcal L}_\beta\to{\mathcal L}_{\alpha+\beta}\) is an isomorphism for every \(\alpha,\beta\in\mathop{\text{Pic}}(X_\Delta)_{\geq 0}\). In the last section of the paper the author applies this result in the case \(Z=X_\Delta\) and deduces the previous results mentioned above, in most cases in a stronger form than previously. He also gives some examples. Cartier divisor; coordinate ring; morphisms into toric varieties; topology of torus actions Kajiwara, T.: The functor of a toric variety with enough invariant effective cartier divisors. Tôhoku math. J. 50, 139-157 (1998) Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Birational geometry The functor of a toric variety with enough invariant effective Cartier divisors

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties In the paper under review, homogeneous vector bundles \(E\) on \(\mathbb P^2= \mathbb P(V)\) are studied by means of their minimal free resolution \[ 0 \to \bigoplus_q {\mathcal O}(-q) \otimes A_q \to \bigoplus_q {\mathcal O}(-q) \otimes B_q \to E \to 0 \] with SL\((V)\)-invariant maps, and \(A_q,B_q\) being SL\((V)\) representations. The representations and the maps \(\bigoplus_q {\mathcal O}(-q) \otimes A_q \to \bigoplus_q {\mathcal O}(-q) \otimes B_q\) which can occur are completely characterized. Then, conditions for \(E\) to be simple or stable are studied in terms of the above resolution when the first term is irreducible and the second one is regular. Infinitely many examples of unstable simple homogeneous bundles are also given. minimal resolutions; quivers Ottaviani G., Rubei E.: Resolutions of homogeneous bundles on P 2. Ann. Inst. Fourier Grenoble 55(3), 973--1015 (2005) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Homogeneous spaces and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets Resolutions of homogeneous bundles on \({\mathbb P}^2\).

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(G\) be a semisimple, simply connected, algebraic group over an algebraically closed field \(k\) with Lie algebra \(\mathfrak{g}\). We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of \(\mathfrak{g}\otimes k((\pi))\), i.e. fixed point varieties on affine flag manifolds. We define a natural class of \(k^*\)-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair \((N,f)\) consisting of \(N\in \mathfrak{g}\otimes k((\pi))\) and a \(k^*\)-action \(f\) of the specified type which guarantees that \(f\) induces an action on the variety of parahoric subalgebras containing \(N\). For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the \(k^*\)-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of \(\mathfrak{g}\). --. --. --. --., The geometry of fixed point varieties on affine flag manifolds , Trans. Amer. Math. Soc. 352 (2000), 2087--2119. JSTOR: Group actions on varieties or schemes (quotients), Linear algebraic groups over local fields and their integers The geometry of fixed point varieties on affine flag manifolds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Fix integers \(n,g,d\) with \(n\geq 3\), \(g\geq 2\) and \(d\geq ng+1\). Let \(C\) be a smooth projective curve of genus \(g\) and \(L\in\text{Pic}(C)\) with \(\deg(L)=d\). Let \(f\colon C\to\mathbb{P}^n\) be a general embedding of \(C\) with \(f^*(\mathcal O_{P^n}(1))\cong L\). Here we prove that a general finite subset \(S\) of \(f(C)\) with large degree satisfies the minimal resolution conjecture, i.e. for every integer \(i\) with \(1\leq i\leq n\) at most one of the Betti numbers \(b_{i,j}(S)\), \(j\in Z\), of \(S\) is non-zero. Syzygies, resolutions, complexes and commutative rings, Curves in algebraic geometry, Projective techniques in algebraic geometry On the minimal free resolution of finite subsets of projective curves.

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties How is the resolution of the ideal of a set of distinct generic points in \({\mathbb{P}}^n\) like? A conjecture about the graded Betti numbers of such resolutions (known as the ``minimal resolution conjecture'', MRC) was given by \textit{A. Lorenzini} [J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)], and it has been proved in many cases and even asymptotically (when the number of points is much bigger then \(n\), by a result of Hirschowitz and Simpson). Examples found computationally (Schreyer 1993) suggested nevertheless that the MRC could be false in general, even if no ``geometrical reason'' for those (three) counterexamples was known. In the beautiful paper under review, such reason is found, enclosed in the theory of ``Gale transforms'' (they could be viewed as duality of linear series on a finite Gorenstein scheme), a way to associate to a set \(\Gamma \) of \(\gamma \geq r+3\) distinct points in \({\mathbb{P}}^r\) another set \(\Gamma ^\prime \) of \(\gamma \) points in \({\mathbb{P}}^s\), with \(s={\gamma -r -2}\). The idea, expressed in classical language is this: With an appropriate choice of the coordinates, we can suppose that the coordinates of the points of \(\Gamma \) are the rows of a \((r+1)\times \gamma\) matrix \((I_{r+1}|B)\); then the coordinates of the points in \(\Gamma ^\prime \) are the rows of the matrix \((B^T|I_{s+1})\). In the paper the relation among graded Betti numbers of \(\Gamma \) and of \(\Gamma ^\prime\) are found, and the ``mystery'' of the counterexamples to MRC is solved, moreover an infinite family of counterexample is determined (in any \({\mathbb{P}}^r\) with \(r\geq 6\), \(r\neq 9\)). Gale duality; generic points; minimal resolution conjecture Eisenbud, D.; Popescu, S., Gale duality and free resolutions of ideals of points, Invent. Math., 136, 419-449, (1999) Syzygies, resolutions, complexes and commutative rings, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Gale duality and free resolutions of ideals of points

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties For most projective surfaces \(S\) of general type, the map \(\varphi_{2K}\) defined by the bicanonical linear series \(|2K|\) is birational. There are exceptions, like surfaces containing a pencil of curves of genus \(2\), which are called ``standard examples''. A classification of surfaces of general type with \(\varphi_{2K}\) non birational is not yet completely known. The authors and P. Francia classified all non standard examples with geometric genus \(p_g\) at least \(4\), while the classification of examples with \(p_g=3\) and irregularity \(q>0\) was achieved by the authors and F. Catanese. The authors complete here the classification of surfaces of general type with \(p_g=3\) and \(\varphi_{2K}\) non birational, taking care of the regular case. It turns out that the only non-standard exceptions are either a specialization of irregular exceptions, or double planes branched along octic curves, or surfaces with \(K^2=8\), described in the paper. The classification is achieved by a careful examination of the rational map induced by the canonical series \(|K|\), which turns out to be not composed with a pencil in all non-standard examples. non-birational bicanonical map; canonical linear series; surfaces of general type Ciro Ciliberto and Margarida Mendes Lopes, On regular surfaces of general type with \?_{\?}=3 and non-birational bicanonical map, J. Math. Kyoto Univ. 40 (2000), no. 1, 79 -- 117. Surfaces of general type, Divisors, linear systems, invertible sheaves On regular surfaces of general type with \(p_g=3\) and non-birational bicanonical map

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Consider a system of \(n\) polynomial equations in \(n\) variables over an algebraically closed field \(K\), where the occurring monomials are fixed and the coefficients are ``generic''. In this paper formulae for the number of roots of such a system in any \(K^*\)-stable subset of the affine space \(K^n\) are given. There are two main results, a recursion over ``shifted'' and lower-dimensional systems (main theorem I), and an expression for a special situation in terms of mixed volumes (affine point theorem II). The paper also contains a characterization of ``genericity'' of the coefficients in terms of sparse resultants (main theorem II). Moreover, for the case of affine space minus an arbitrary union of coordinate hyperplanes, the formulae yield upper bounds on the number of possible isolated roots. With some examples, the author illustrates that those bounds are sharper than the ones obtained from previously known results. There are two main new ideas, one is to construct from the Newton-polytopes of the given system a suitable projective toric compactification for \(K^n\), and the other is to consider ``shifts'' of the modified polytopes to take into account roots in certain coordinate subspaces. counting solutions of polynomial equations; mixed volumes Rojas, J. M.: Toric intersection theory for affine root counting. J. pure appl. Algebra 136, 67-100 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Mixed volumes and related topics in convex geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Numerical computation of solutions to systems of equations Toric intersection theory for affine root counting

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(N\geq 1\) be an integer. Let \(X_ 0(N)\) be the modular curve over \({\mathbb{Z}}\), as constructed by Katz and Mazur. The minimal resolution of \(X_ 0(N)\) over \({\mathbb{Z}}[1/6]\) is computed. Let \(p\geq 5\) be a prime, such that \(N=p^ 2M\), with M prime to p. Let \(n=(p^ 2-1)/2\). It is shown that \(X_ 0(N)\) has stable reduction at p over \({\mathbb{Q}}[^ n\sqrt{p}]\), and the fibre at p of the stable model is computed. modular curve; minimal resolution; stable reduction Edixhoven, B, Minimal resolution and stable reduction of \(X_0(N)\), Ann. Inst. Fourier, 40, 31-67, (1990) Global ground fields in algebraic geometry, Arithmetic ground fields for curves Minimal resolution and stable reduction of \(X_ 0(N)\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties For an algebraic variety \(X\) over an algebraically closed field \({\mathbb K}\) of arbitrary characteristic, a sheaf \({\mathcal F}\) on \(X\) is called \textit{immaculate} if all cohomology groups \(H^p(X,{\mathcal F})=0\) for all \(p\in{\mathbb Z}\). The main focus of the paper under review is the structure of the family of all immaculate line bundles on \(X\) as a subset of the group \(\text{Pic}(X)\). Classically, the cohomology of a Weil divisor on a toric variety is calculated using polyhedra complexes contained in \(N_{\mathbb R}\), where \(N\) is the lattice of \(1\)-parameter subgroups of the torus acting on \(X\). The first contribution of the paper under review is a shift of the classical approach, viewing now the cohomology of a toric \({\mathbb Q}\)-Cartier Weil divisor using polytopes in the space \(M_{\mathbb R}\), where \(M\) is the dual lattice of \(N\). For projective toric varieties, Theorem 3.6 describes the \(M\)-graded cohomology groups \(H^i(X,{\mathcal O}(D))\) in terms of the polyhedra associated to a decomposition of the divisor \(D\) as the difference \(D^+-D^-\) of two nef divisors. For the main objective, a description of the locus of all immaculate line bundles in the class group \(\text{Pic}(X)\) of a toric variety \(X\), the first results establish some general invariance properties of immaculacy (or a relative version of it) of locally free sheaves under various types of morphisms between toric varieties. Next, to describe the immaculate locus the authors use the map \(\pi:{\mathbb Z}^{\Sigma(1)}\to \text{Pic}(X)\) that assigns to a \(T\)-invariant divisor its class. Using this map, the first task is to identify the \(T\)-invariant divisors whose images carry some cohomology by using an approach similar to the one used for acyclic line bundles as in [\textit{L. Borisov} and \textit{Z. Hua}, Adv. Math. 221, No. 1, 277--301 (2009; Zbl 1210.14006)] and [\textit{A. I. Efimov}, J. Lond. Math. Soc., II. Ser. 90, No. 2, 350--372 (2014; Zbl 1318.14047)]. In Section 5 of the paper under review the authors identify some subsets of \(\Sigma(1)\) whose images under \(\pi\) either carry some cohomology or not. One of the main results, Theorem 5.24, essentially describes the locus of immaculate line bundles for a complete simplicial toric variety. Moreover, in some concrete instances the conditions on the subsets of \(\Sigma(1)\) can be used to describe the locus of immaculate bundles, for example for smooth projective toric varieties of Picard rank \(2\) in Theorem 6.2 . Using the classification of smooth projective toric varieties of Picard rank \(3\) of \textit{V. L. Batyrev} [Tôhoku Math. J., II. Ser. 43, No. 4, 569--585 (1991; Zbl 0792.14026)] in Section 8 the authors consider this situation in two cases, depending on the splitting of the fan of the toric variety. toric variety; immaculate line bundle; splitting fan; Picard rank; primitive collection Toric varieties, Newton polyhedra, Okounkov bodies, Sheaves in algebraic geometry, Vanishing theorems in algebraic geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Immaculate line bundles on toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This paper is concerned with extensions of toric varieties. Let \(K\) be an algebraically closed field of any characteristic, and let \(S\) be a subsemigroup of \(\mathbb{N}^d\) generated by \({\mathbf m}_1,\ldots,{\mathbf m}_n\). Denote by \(S_{l,{\mathbf m}}\) the affine semigroup generated by \(l{\mathbf m}_1,\ldots,l{\mathbf m}_n\) and \({\mathbf m} \in \mathbb{N}^d\), where \(l\) is a positive integer. The affine toric variety \(V_{S_{l,{\mathbf m}}} \subset \mathbb{A}^{n+1}\) is an extension of \(V_{S} \subset \mathbb{A}^{n}\), if \({\mathbf m} \in S\), and \(l\) is a positive integer relatively prime to a component of \({\mathbf m}\). A projective variety \(\overline{E} \subset \mathbb{P}^{n+1}\) is an extension of another one \(\overline{X} \subset \mathbb{P}^{n}\) if its affine part \(E\) is an extension of the affine part \(X\) of \(\overline{X}\). The author proves that affine extensions can be obtained by gluing semigroups, and therefore their minimal generating sets can be obtained by adding a binomial. In the projective case a similar result holds under a mild condition. Starting with a set-theoretic complete intersection, arithmetically Cohen-Macaulay or Gorenstein toric variety, he obtains infinitely many toric varieties having the same property. It was conjectured by M. E. Rossi that the Hilbert function of any Gorenstein local ring is non-decreasing. The author shows that if a toric variety has a Cohen-Macaulay tangent cone or at least its local ring has a non-decreasing Hilbert function, then its nice extensions share these properties, supporting Rossi's conjecture for higher dimensional Gorenstein local rings. toric varieties; affine extensions; Gorenstein local rings Şahin, M, Extensions of toric varieties, Electron. J. Combin., 18, p93, (2011) Toric varieties, Newton polyhedra, Okounkov bodies, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Complete intersections, Syzygies, resolutions, complexes and commutative rings Extensions of toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a reduced complex space and \(r \colon \widetilde X \to X\) a resolution of singularities. If \(\alpha\) is a holomorphic differential form on the smooth locus \(X_{\mathrm{reg}}\), we can pull it back to \(r^{-1}(X_{\mathrm{reg}})\) and ask whether the pulled back form extends holomorphically to all of \(\widetilde X\). This is the ``extension problem'' studied in this paper. While it seems technical at first sight, knowing the extension property for certain classes of singularities is actually very useful. The best result available so far is by \textit{D. Greb} et al. [Publ. Math., Inst. Hautes Étud. Sci. 114, 87--169 (2011; Zbl 1258.14021)] and says that if \(X\) is algebraic and has only klt singularities, then \(p\)-forms extend in the above sense, for any value of \(0 \le p \le n = \dim X\). Philosophically (if \(X\) is Gorenstein, this is even technically correct) this can be rephrased as saying that ``if \(n\)-forms extend, then \(p\)-forms extend for all \(p \le n\)''. This epic paper actually proves the latter statement, and generalizations of it. {Theorem~1.4.} In the above situation, if \(k\)-forms extend for some \(0 \le k \le n\), then \(p\)-forms extend for all \(p \le k\). Theorem~1.5 is a version where logarithmic poles along the exception divisor of \(r\) are allowed in the extension process. Several applications of these new results are given. The other main improvement over [loc. cit.] (besides replacing the klt assumption by a weaker, more natural condition) is that \(X\) is no longer assumed to be algebraic. For people working with e.g.~singular Kähler spaces, this is extremely useful. It is however conceivable that due to very recent results of \textit{O. Fujino} [``Minimal model program for projective morphisms between complex analytic spaces'', Preprint. \url{arXiv:2201.11315}], the original proof of \textit{D. Greb} et al. [Publ. Math., Inst. Hautes Étud. Sci. 114, 87--169 (2011; Zbl 1258.14021)] can be carried over almost verbatim to complex spaces (only in the klt case, of course). extension theorem; holomorphic form; complex space; mixed Hodge module; decomposition theorem; rational singularities; pull-back Singularities in algebraic geometry, Local cohomology and algebraic geometry, Global theory of complex singularities; cohomological properties Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let a linear algebraic group \(G\) act on an algebraic variety \(X\). Classification of all these actions, in particular birational classification, is of great interest. A complete classification related to Galois cohomologies of the group \(G\) was established. Another important question is reducibility, in some sense, of this action to an action of \(G\) on an affine variety. It has been shown that if the stabilizer of a typical point under the action of a reductive group \(G\) on a variety \(X\) is reductive, then \(X\) is birationally isomorphic to an affine variety \( \bar X \) with stable action of \(G\). In this paper, I show that if a typical orbit of the action of \(G\) is quasiaffine, then the variety \(X\) is birationally isomorphic to an affine variety \( \bar X \). Group actions on varieties or schemes (quotients), Rational and birational maps, Group actions on affine varieties Varieties birationally isomorphic to affine \(G\)-varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties [For the entire collection see Zbl 0607.00005.] Let \(f(z_ 0,...,z_ n)\) be a germ of an analytic function with an isolated critical point at the origin; f is assumed to have a non- degenerate Newton boundary \(\Gamma\) (f). Let V be the germ of the hypersurface \(f^{-1}(0)\) at the origin. There is always a canonical resolution \(\pi: \tilde V\to V\) of V which is associated with a given simplicial subdivision \(\Sigma^*\) of the dual Newton diagram \(\Gamma^*(f)\). The main purpose of this paper (see in particular {\S} 5) is to study the topology of the exceptional divisors E(P) through a canonical simplicial subdivision \(\Sigma^*\), which is constructed in {\S} 3. If P is a strictly positive vertex of \(\Sigma^*\), then E(P) is always a compact divisor such that \(\pi (E(P))=\{0\}\). The topology of exceptional divisors E(P) of the two dimensional and the three dimensional singuarities is then studied in detail in {\S}{\S} 6 and 8 respectively. In section \(7\) the author shows that the fundamental group of E(P), where P is a strictly positive vertex of a fixed subdivision \(\Sigma^*\), is a finite cyclic group (with an order independent of the choice of \(\Sigma^*)\) if \(n>2\) and if the face \(\Delta\) (P) of \(\Gamma\) (f) is an n-simplex. In section \(9\) the canonical divisors of \(\tilde V\) and E(P) are considered. In particular, applying (9.1) and (9.2) one can calculate the signature of the Milnor fibre of f in the case \(n=2\) from the Newton boundary \(\Gamma\) (f). resolution of hypersurface singularities; simplicial subdivision; topology of the exceptional divisors; canonical; fundamental group; signature of the Milnor fibre; Newton boundary Oka (M.).- On the resolution of the hypersurface singularities. In Complex analytic singularities, volume 8 of Adv. Stud. Pure Math., pages 405-436. North-Holland, Amsterdam (1987). Zbl0622.14012 MR894303 Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Topological properties in algebraic geometry On the resolution of the hypersurface singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The paper is devoted to the determination of the minimum number of equations needed to define set-theoretically an algebraic variety \(V\) (the arithmetical rank of \(I=I(V)\), denoted by ara\((I))\). If \(h(I)=\text{ara}(I)\) (where \(h(I)\) denotes as usual the height of \(I)\) the ideal \(I\) is called a set-theoretic complete intersection, and when ara\((I)\leq h(I)+1\) it is called almost set-theoretically complete intersection. The authors prove that every simplicial toric affine or projective variety with full parametrization is an almost set-theoretically complete intersection; the same is true for every simplicial toric (affine or projective) variety of codimension two. If the characteristic is positive then every simplicial toric affine or projective variety with full parametrization is actually a set-theoretically complete intersection. set-theoretic complete intersection; arithmetical rank DOI: 10.1006/jabr.1999.8195 Toric varieties, Newton polyhedra, Okounkov bodies, Complete intersections On simplicial toric varieties which are set-theoretic complete intersections

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties In this lecture, the author explains how the complete intersection property is interpreted in terms of \(K\)-theory and/or Chow groups of zero cycles. He starts with the fact that if \(A\) is a Dedekind domain, the following three properties are equivalent: (i) \(A\) has the unique factorization property, (ii) every maximal ideal of \(A\) is a principal ideal, (iii) the divisor class group of \(A\) is trivial. Thereafter, in order to deal with the higher dimensional cases, he leads us to the notions of Picard varieties, Chow groups, K-theory, and describes an intimate connection between the complete intersection property and two conjectures due to Bloch, and Bloch-Beilinson. Several related results of his own in [\textit{A. Krishna} and \textit{V. Srinivas}, London Mathematical Society Lecture Note Series 343, 264--277 (2007; Zbl 1126.14011)] and of some others are discussed too. algebraic cycles; Picard group; Chow group; zero cycles Algebraic cycles, Complete intersections Zero cycles and complete intersection points on affine varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author proves that if \(X\) is a smooth surface in the affine \(n\)-space \((n \geq 5)\), satisfying the following numerical condition: \(c^ 2_ 1(X) + c_ 2(X) = 0\), then \(X\) is a set-theoretic complete intersection. As previous results in the same direction [see \textit{N. Mohan Kumar}, Mém. Soc. Math. Fr., Nouv. Sér. 38, 135-143 (1989; Zbl 0715.14043); and \textit{S. Bloch}], the proof makes use of the Ferrand construction. Moreover, the following example is constructed: there exists a smooth affine fourfold \(X\) over \(\mathbb{C}\) such that \(\omega_ X\) is trivial and \(\Omega_ X\) is not stably trivial (hence \(X\) is not a complete intersection in an affine space). Previously known examples of this phenomenon have dimension at least five. Chern class; complete intersection Hauber, P., Smooth affine varieties and complete intersections, Manuscripta Math., 83, 265-277, (1994) Complete intersections, Characteristic classes and numbers in differential topology Smooth affine varieties and complete intersections

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author considers the toric birational morphisms between toric varieties of dimension \(3\) which contract m irreducible divisors to a point. In particular the author studies the question whether they are factorizable in a product of blowings-up of points or not and gives a complete classification for the case \(m\leq 5\), using combinatorial methods. The most important tool for the proof is the ``Farey graph'' associated to such a morphism. toroidal embeddings of 3-folds; toric birational morphisms; product of blowings-up; Farey graph Mina Teicher, On toroidal embeddings of 3-folds, Israel J. Math. 57 (1987), no. 1, 49 -- 67. Rational and birational maps, \(3\)-folds On toroidal embeddings of 3-folds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes \({\mathcal M}_F\) of polynomial matrices. Let \(X\) be the algebraic curve given by the common characteristic equation for \({\mathcal M}_F\). We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of \(X\). This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As an application, we discuss the algebraic complete integrability of the extended Lotka-Volterra lattice with a periodic boundary condition. linearization; completely integrable Hamiltonian systems; polynomial matrices Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Lattice dynamics; integrable lattice equations, Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Discrete version of topics in analysis The matrix realization of affine Jacobi varieties and the extended Lotka--Volterra lattice

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups. orbifold; symplectic; isotropy; resolution Symplectic manifolds (general theory), Topology and geometry of orbifolds, Global theory and resolution of singularities (algebro-geometric aspects) Symplectic resolution of orbifolds with homogeneous isotropy

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of general type. Rigidity of surjective morphisms, and the failure of a morphism to be rigid have been studied by a numher of authors in the past. The main result of this paper states that surjective morphisms are always rigid, unless there is a clear geometric reason for it. More precisely, we can say the following. First, deformations of surjective morphisms between normal projective varieties are unobstructed unless the target variety is covered by rational curves. Second, if the target is not covered hy rational curves, then surjective morphisms are infinitesimally rigid, unless the morphism factors via a variety with positive-dimensional automorphism group. In this case, the Hom-scheme can be completely described. Hwang, J.-M., Kebekus, S., Peternell, T.: Holomorphic maps onto varieties of non-negative Kodaira dimension. J. Algebr. Geom. 15, 551--561 (2006) Birational automorphisms, Cremona group and generalizations, Varieties and morphisms, Automorphisms of surfaces and higher-dimensional varieties, \(n\)-folds (\(n>4\)), Rational and birational maps, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables Holomorphic maps onto varieties of non-negative Kodaira dimension

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Nous étudions les polynômes \(F\in\mathbb{C} \{S_\tau\} [Y]\) à coefficients dans l'anneau de germes de fonctions holomorphes au point spécial d'une variété torique affine. Nous généralisons à ce cas la paramétrisation classique des singularités quasi-ordinaires. Cela fait intervenir d'une part une généralisation de l'algorithme de Newton-Puiseux, et d'autre part une relation entre le polyèdre de Newton du discriminant de \(F\) par rapport à \(Y\) et celui de \(F\) au moyen du polytope-fibre de \textit{L. I. Billera} et \textit{B. Sturmfels} [Ann. Math., II. Ser. 135, 527-549 (1992; Zbl 0762.52003)]. Cela nous permet enfin de calculer, sous des hypothèses de non dégénérescence, les sommets du polyèdre de Newton du discriminant à partir de celui de \(F\), et les coefficients correspondants à partir des coefficients des exposants de \(F\) qui sont dans les arêtes de son polyèdre de Newton. Newton polyhedron; germs; affine toric variety; quasi-ordinary singularities; discriminant González Pérez, P.D., Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canad. J. math., 52, 2, 348-368, (2000) Toric varieties, Newton polyhedra, Okounkov bodies, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Germs of analytic sets, local parametrization Quasi ordinary toric singularities and Newton polyhedron of the discriminant

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Fix integers \(s>0\), \(m_1>\cdots>m_s>0\), \(a_i>0\), \(1\leq i\leq s\). Take \(\sum^s_{i=1}a_i\), general points \(P_{i,j}\in\mathbb{P}^2\), \(1\leq i\leq s\), \(1\leq j\leq a_i\) and set \(Z:=\bigcup^s_{i=1}\bigcup^{a_i}_{j=1}m_i P_{i,j}\subset\mathbb{P}^2\) and \(\delta:=\sum^s_{i=1}a_i{m_i\choose 2}\). Let \(k\) be the minimal integer such that \(\delta\leq{k+2\choose 2}\). Assume \(m_s=1\) and either \(a_s\geq km(m_1+2) (m_1+1)/2+m_1k\) or \(m_{s-1}=2\), \(m_s\geq k\) and \(2a_{s-1}+a_s\geq km(m_1+2) (m_1+ 1)/2+m_1k\) or char\((\mathbb{K})\neq 2\), \(m_{s-1}=2\), \(2a_{s-1}+a_s\geq km(m_1+2) (m_1+1)/2+m_1k\) and \(m_s\geq 2\). Then \(h^0(\mathbb{P}^2, {\mathcal I}_Z(t))=0\) for all \(t<k\), \(h^1(\mathbb{P}^2,{\mathcal I}_Z(t))=0\) for all \(t\geq k\) and the homogeneous ideal of \(Z\) is minimally generated by \({k+2\choose 2}-\delta\) forms of degree \(k\) and by \(\max\{0,2\delta-k(k+2)\}\) forms of degree \(k+1\). Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings On the minimal free resolution of general unions of fat points in \(\mathbb{P}^2\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We examine embeddings of minimal abelian surfaces \(A\) into smooth toric 4-folds \(X\) of Picard number 2. We show that for most \(X\) no such embeddings exist. For one particular \(X\), however, we exhibit a 2-dimensional family of abelian surfaces \(A\) embedded in \(X\), and hence a rank 2 vector bundle on \(X\). We use a simple description of morphisms from normal varieties into smooth toric varieties (based on joint work with T. Oda) to construct the embeddings. embeddings of minimal abelian surfaces; smooth toric 4-folds Embeddings in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Low codimension problems in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(4\)-folds, Abelian varieties and schemes Abelian surfaces in toric 4-folds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(\Gamma\subset\text{SL}_2(\mathbb{C})\) be a finite nontrivial subgroup. Associated to \(\Gamma\) is a Kleinian singularity \(X:=\mathbb{C}^2/\Gamma\). \textit{W. Crawley-Boevey} and \textit{M. P. Holland} [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)] introduced a family \(\{\mathcal O^\lambda\}\) of (generally) noncommutative deformations of \(X\). The main result of this paper is the construction for certain \(\lambda\) with \(\mathcal O^\lambda\) having finite global dimension of a family of filtered \(\mathbb{Z}\)-algebras \(\{B^\lambda(\chi)\}\) with each \(B^\lambda(\chi)\) being Morita equivalent to \(\mathcal O^\lambda\). This is an analogue of a result of \textit{I. Gordon} and \textit{J. T. Stafford} [Adv. Math. 198, No. 1, 222-274 (2005; Zbl 1084.14005)] which states that certain rational Cherednik algebras are Morita equivalent to certain noncommutative deformations of a Hilbert scheme. Both of these results were conjectured by V. Ginzburg. This result provides a new approach to studying the representation theory of \(\mathcal O^\lambda\) which the author investigates in forthcoming work. Moreover, the associated graded \(\mathbb{Z}\)-algebra of \(B^\lambda(\chi)\) is shown to be Morita equivalent to the minimal resolution of \(X\). This affirmatively answers a question posed by \textit{M. P. Holland} [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 6, 813-834 (1999; Zbl 1036.16024)]. In the sense of Holland's work, the algebra \(\mathcal O^\lambda\) can be considered as a `quantization' of \(X\), and Holland asked whether the minimal resolution of \(X\) admitted a quantization. This result shows that these graded algebras are such quantizations. The paper includes a discussion of quivers, particularly the McKay quiver of \(\Gamma\) which is needed for the definition of \(\mathcal O^\lambda\), as well as of the construction of minimal resolutions of \(X\). The author also discusses some fundamental notions about \(\mathbb{Z}\)-algebras and introduces the concept of a Morita \(\mathbb{Z}\)-algebra. A key step in the proof is a stronger version of a categorical equivalence result of Gordon and Stafford. It should be noted that using independent methods a slightly different version of the main result here has been proved by \textit{I. M. Musson} [J. Algebra 293, No. 1, 102-129 (2005; Zbl 1082.14008)] for Kleinian singularities of type \(A\). Kleinian singularities; noncommutative deformations; minimal resolution; quantizations; \(\mathbb{Z}\)-algebras M. Boyarchenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math., 211 (2007), 244--265. Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Deformations of singularities, Geometric invariant theory, Representations of associative Artinian rings, Module categories in associative algebras, Deformations of associative rings, Modifications; resolution of singularities (complex-analytic aspects), Graded rings and modules (associative rings and algebras) Quantization of minimal resolutions of Kleinian singularities.

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(\mathbb{C}\) be the base field. Let \(G\) be a reductive linear algebraic group acting properly on a smooth projective variety \(X\). By Geometric Invariant Theory, one has open subsets \(X^s \subset X^{ss} \subset X\) of stable and semi-stable points. The quotient \(X^{ss}/G\) is projective and generally singular; and the quotient \(X^{s}/G\) has at most finite quotient singularities. \textit{F. C. Kirwan} [Ann. Math. (2) 122, 41--85 (1985; Zbl 0592.14011)] described a procedure of blowing \(X\) up along a sequence of smooth \(G\)-invariant subvarieties to obtain a variety \(X'\) with a \(G\)-action, such that every semistable point of \(X'\) is stable. Hence, the quotient variety \((X')^{ss}/G\) is projective with only finite quotient singularities, and there is an induced projective birational morphism \((X')^{ss}/G \to X^{ss}/G\) which is an isomorphism over the open set \(X^s/G\). One can therefore consider \((X')^{ss}/G\) as a partial desingularization of \(X^{ss}/G\). In this paper, the authors study a similar construction when \(\mathcal{X}\) is an Artin toric stack. To do this, they use \textit{Reichstein transformations}, which are certain birational transformations of Artin stacks with good moduli spaces. Let \(\mathcal{C} \subset \mathcal{X}\) be a closed substack. Then the Reichstein transformation of \(\mathcal{X}\) relative to \(\mathcal{C}\) is defined to be the complement of the strict transform of the saturation of \(\mathcal{C}\) relative to the quotient map \(q: \mathcal{X} \to M\) in the blow-up of \(\mathcal{X}\) along \(\mathcal{C}\). Theorem 4.7 states that the Reichstein transformation of a toric stack along a toric substack is another toric stack, which can moreover be described combinatorially in terms of the original toric stack. Applying this result repeatedly gives the result. Edidin, D.; More, Y., Partial desingularizations of good moduli spaces of Artin toric stacks, Michigan Math. J., 61, 451-474, (2012) Stacks and moduli problems, Toric varieties, Newton polyhedra, Okounkov bodies Partial desingularizations of good moduli spaces of Artin toric stacks

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(K\) be an algebraically closed field and \(K^\ast=K\smallsetminus \{0\}\). Let \(A=[a_0, \dots, a_{n+1}]\) be an \(n\times (n+2)\) integer matrix whose columns span the lattice \(\mathbb{Z}^n,\;a_i=(a_{ji})\). Let \(t^{a_i}=t_1^{a_{1i}}\cdot\dots\cdot t_n^{a_{ni}}\) and consider the map \(\Phi_A:(K^\ast)^n\to (K^\ast)^{n+2}\) defined by \(\Phi_A(t)=(t^{a_0}, \dots, t^{a_{n+1}})\). Assume that all columns of \(A\) sum to the same positive integer \(d\). Then \(\Phi_A\) induces as a map \((K^\ast)^n\to \mathbb{P}^{n+1}\). The toric variety \(X_A\) is the closure in \(\mathbb{P}^{n+1}\) of the image of \(\Phi_A\). Let \(f=(f_0(x), f_1(x), \dots, f_{n+1}(x))\in K[x]^{n+2}\) be a vector of univariate polynomials in the variable \(x\). This vector defines a parametric curve \(Y_f\subset \mathbb{P}^{n+1}\), the closure of the set \(\{(f_0(x):\dots:f_{n+1}(x))\}\). Let \(Z_{A, f}\) be the closure of the set \[ \{(t^{a_0} f_0(x):t^{a_1} f_1(x):\dots : t^{a_{n+1}} f_{n+1}(x)\;|\;f\in (K^\ast)^n, x\in K\}\subseteq \mathbb{P}^{n+1}\;, \] the so--called Hadamard product of \(X_A\) and \(Y_f\). The Plücker matrix associated with \(A\) is the \((n+2)\times(n+2)\) matrix \[ P_A=(p_{ij}) , \;p_{ij}=\begin{cases} \frac{1}{\delta}(-1)^{i+j}\det(A_{[i,j]} )& i<j\\ -p_{ji} & i>j\\ 0 & i=j. \end{cases} \] Here \(\delta\) is the greatest common divisor of all \(\det(A_{[i,j]})\) and \(A_{[i,j]}\) is the \(n\times N\) submatrix of \(A\) obtained by deleting the columns \(a_i\) and \(a_j\). The valuation matrix associated with \(f\) is an integer matrix \(V_f\) with \(n+2\) rows defined as follows. Let \(g_1, \dots, g_m\) be all distinct linear factors of \(f_0, \dots, f_{n+1}\) and let \(\mathrm{ord}_{g_j} f_i\) be the multiplicity of \(g_j\) in \(f_i\). Let \(u_j:=(\mathrm{ord}_{g_j} f_0, \dots, \mathrm{ord}_{g_j} f_{n+1})\) and \(S=\{u_1, \dots, u_m\}\). If two vectors in \(S\) are linearly dependent, then we delete them and add their sum to the set. After finitely many steps we end up with \(S'=\{v_1, \dots, v_l\}\). Now \(V_f:=[v_1^T v_2^T\cdots v_l^T(-\sum\limits_{j=1}^l v_j)^T]\). The following theorem is proved. If \(\mathrm{rank}(P_A\cdot V_f)=0\) then \(Z_{A,f}\) is not a hypersurface. If \(\mathrm{rank}(P_A\cdot V_f)=1\) then \(Z_{A,f}\) is a toric hypersurface. If \(\mathrm{rank}(P_A\cdot V_f)=2\) then \(Z_{A,f}\) is a hypersurface but not toric. In case \(Z_{A,f}\) is a hypersurface, it is called the almost toric hypersurface associated to \((A, f)\). The Newton polytope of \(Z_{A,f}\) is described and an algorithm to compute a defining equation for \(Z_{A,f}\). toric variety; tropical geometry; almost toric Hypersurfaces and algebraic geometry, , Toric varieties, Newton polyhedra, Okounkov bodies Almost-toric hypersurfaces

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a normal projective \(\mathbb{Q}\)-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in \(X\) via the total stringy Chern class of \(X\). This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective \(\mathbb{Q}\)-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating \(d\)-dimensional reflexive polytopes to the number 12 in dimension \(d \geq 4\). Mirror symmetry (algebro-geometric aspects), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Toric varieties, Newton polyhedra, Okounkov bodies, Complete intersections, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Stringy Chern classes of singular toric varieties and their applications

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The goal of the paper under review is ``to give some evidence that the fundamental groups of normal varieties behave like those of smooth varieties''. In particular, the authors generalize a main theorem from [\textit{D. Arapura}, J. Algebr. Geom. 6, No. 3, 563--597 (1997; Zbl 0923.14010)] and describe the jump loci phenomenon of rank one local systems on an irreducible normal variety \(X\) using the resonance and characteristic varieties of \(X\) and its resolution [\textit{A. Dimca} et al., Duke Math. J. 148, No. 3, 405--457 (2009; Zbl 1222.14035)]. They also give a detail proof of the well-known Deligne's statement that Morgan's theorem holds for normal varieties, obtain an analog of Theorem 11.7 [loc. cit.] concerning conditions under which a right-angled Artin group is the fundamental group of a normal variety, etc. In conclusion the authors remark that ``it would be interesting to find an example of a group which is the fundamental group of a normal variety but not the fundamental group of a smooth variety''. normal varieties; fundamental groups; local systems; twisted cohomology; resonance varieties; characteristic varieties; Malcev Lie algebras, Hodge structure Arapura D., Dimca A. and Hain R., On the fundamental groups of normal varieties, Commun. Contemp. Math. 18 (2016), no. 4, Article ID 1550065. Homotopy theory and fundamental groups in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Hodge theory in global analysis, Mixed Hodge theory of singular varieties (complex-analytic aspects), Homology with local coefficients, equivariant cohomology On the fundamental groups of normal varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(P\) be an isolated singularity of an algebraic variety \(V\). The target of the paper is a natural extension of the notion of ordinary singularities of curves to varieties of arbitrary dimension. The authors consider the case where the normalization of \(V\) at \(P\) is regular around the point, a situation for which they provide huge classes of examples. In this case, the authors say that \(P\) is an ordinary singularity of \(V\) when the projectivized tangent cone at \(P\) is reduced. For ordinary singularities, the authors prove that the projectivized tangent cone is the union of \(e\) linear spaces of the same dimension, \(e\) being the multiplicity of \(P\). When the linear spaces that compose the projectivized tangent cone are in general position, the authors prove that also the affine tangent cone at \(P\) turns out to be reduced. singularities Singularities of surfaces or higher-dimensional varieties Ordinary isolated singularities of algebraic varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(k\) be a field and let \(F= \{x^{\alpha_1},\dots, x^{\alpha_m}\}\subset k[x_1,\ldots,x_{n}]\) be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra \(R[F t]\) and the subring \(k[F ]\). If the monomials in \(F\) have the same degree, one of the consequences is a criterion for the \(k\)-rational map \(F:\mathbb{P}^{n-1}_k \to \mathbb{P}^{m-1}_k\) defined by \(F\) to be birational onto its image. birationality of monomial rational map; minors; normal ideal; Rees algebras A. Simis and R. H. Villarreal, Constraints for the normality of monomial subrings and birationality, Proc. Amer. Math. Soc. 131 (2003), 2043--2048. Polynomial rings and ideals; rings of integer-valued polynomials, Rational and birational maps, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Birational automorphisms, Cremona group and generalizations, Integral closure of commutative rings and ideals Constraints for the normality of monomial subrings and birationality

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(k\) be an algebraically closed field of arbitrary characteristic and \(S\) a nonsingular affine algebraic surface defined over \(k\). Let \((V,D)\) be a pair of a nonsingular projective surface \(V\) and a reduced normal crossing divisor \(D\) on \(V\). We call \((V,D)\) a normal algebraic compactification of \(S\) if \(S\) is isomorphic to \(V-D\). A normal algebraic compatification \((V,D)\) of \(S\) is said to be minimal if \((E\cdot D-E)\geq 3\) for any \((-1)\)-curve \(E\subset D\). Note that minimal normal algebraic compactifications of \(S\) exist since \(S\) is an affine algebraic surface. The author lists all minimal normal algebraic compactifications of \(\mathbb{A}^1_k \times\mathbb{A}^1_{(r)}\), where \(\mathbb{A}^1_{(r)}= \mathbb{A}_k^1-\{r\) points\} \((r\geq 1)\), by using a purely algebraic method. normal compactification; \(\mathbb{P}^1\)-fibration; affine surface Affine fibrations, Fibrations, degenerations in algebraic geometry, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Algebraic compactifications of some affine surfaces.

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author studies affine Deligne-Lusztig varieties \(X_{\tilde{w}}(b)\) in the affine flag variety of a quasi-split tamely ramified group. The term affine refers to the fact that the notion is defined in terms of affine root systems. Let \(G\) be a connected reductive group. For simplicity, suppose \(G\) split over \(\mathbb{F}_q\) and let \(L = k((\epsilon))\) be the field of the Laurent series. The Frobenius automorphism \(\sigma\) on \(G\) induces an automorphism \(\sigma\) on the loop group \(G(L)\). Let \(I\) be a \(\sigma\)-stable Iwahori subgroup of \(G(L)\). By definition, the affine Deligne-Lusztig variety associated with \(\tilde{w}\) in the extended affine Weyl group \(\tilde{W}\cong I \backslash G(L)/I\) and \(b \in G(L)\) is \[ X_{\tilde{w}}(b) = \{gI\in G(L)/I \;|\;g^{ -1} b\sigma(g)\in I\dot{\tilde{w}}I \}, \] where \(\dot{\tilde{w}}\in G(L)\) is a representative of \(\tilde{w}\in \tilde{W}\). Understanding the emptiness/nonemptiness pattern and dimension of affine Deligne-Lusztig varieties is fundamental to understand certain aspects of Shimura varieties with Iwahori level structures. The affine Deligne-Lusztig variety \(X_{\tilde{w}}(b)\) for an arbitrary \(\tilde{w}\in\tilde{W}\) and \(b\in G(L)\) is very difficult to understand. One of the main goal of this paper is to develop a reduction method for studying the geometric and homological properties of \(X_{\tilde{w}}(b)\). In the finite case, Lang's theorem implies that \(G\) is a single \(\sigma\)-conjugacy class. This is the reason why a (classical) Deligne-Lusztig variety depends only on the parameter \(w\in W\), with no need to choose an element \(b\in G\). However, in the affine setting, the analog of Lang's theorem fails and \(X_{\tilde{w}}(b)\) depends on two parameters. Hence it is a challenging task even to describe when \(X_{\tilde{w}}(b)\) is nonempty . To overcome this difficulty, the author proves that Lang's theorem holds ``locally'' for loop groups, using a reduction method. Although the structure of arbitrary affine Deligne-Lusztig varieties is quite complicated, the varieties associated with minimal length elements \(\tilde{w}\in\tilde{W}\) have a very nice geometric structure. The author describes the geometric structure of \(X_{\tilde{w}}(b)\) for such a \(\tilde{w}\) generalizing one of the main results in [\textit{X. He} and \textit{G. Lusztig}, J. Am. Math. Soc. 25, No. 3, 739--757 (2012; Zbl 1252.20047)] to the affine case. The emptiness/nonemptiness pattern and dimension formula for affine Deligne-Lusztig varieties can be obtained keeping track of the reduction step from an arbitrary element to a minimal length element. This is accomplished via the class polynomials of affine Hecke algebras. The affine Deligne-Lusztig variety \(X_{\tilde{w}}(b)\) is nonempty if and only if a certain class polynomial is nonzero. Moreover, the author establishes a connection between the dimension of \(X_{\tilde{w}}(b)\) and the degree of such class polynomial. As a consequence, he proves a conjecture of \textit{U. Görtz} et al. [Compos. Math. 146, No. 5, 1339--1382 (2010; Zbl 1229.14036)]. \(\sigma\)-conjugacy classes; affine Deligne-Lusztig varieties Goertz, U., He, X., Nie, S.: \(P\)-alcoves and non-emptiness of affine Deligne-lusztig varieties, arXiv:1211.3784 Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Linear algebraic groups and related topics Geometric and homological properties of affine Deligne-Lusztig varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(A\) be a commutative ring, and \(S=A[X]\) the polynomial ring over \(A\) in the indeterminates \(X_{ij}\) forming the \(m\times n\) matrix \(X\). Denote the ideal of \(S\) generated by the \(t\)-minors of \(X\) by \(I_ t\). For quite a while it had been an open problem whether the Betti numbers of \(S/I_ t\) are independent of the characteristic of \(A\). The hope for a positive answer was supported by the characteristic free minimal resolutions constructed by Hilbert for \(t=1\) (the Koszul complex), by Eagon and Northcott in the case \(t=\min(m,n)\) and by Akin, Buchsbaum, and Weyman for \(t=\min(m,n)-1\). In the paper under review it is shown that also in the case \(t+2=m=n\) the Betti numbers of \(S/I_ t\) do not depend on the characteristic of \(A\). It follows from this fact that there exists a characteristic free minimal resolution. The authors use the characteristic free representation theory of the general linear group developed by \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 44, 207-278 (1982; Zbl 0497.15020)]. They extend it by providing a `Cauchy formula' for the symmetric algebra of the tensor product of morphisms of finite free modules. (The highly technical details of the article defy an adequate representation in a short review.) In subsequent work \textit{M. Hashimoto} gave an essentially complete answer to the problem of the independence of the Betti numbers; he proved their independence in the more general case in which \(t=\min(m,n)-2\) [J. Algebra 142, No. 2, 456-491 (1991; Zbl 0743.13007)], and showed that in all the cases with \(2<t\leq\min(m,n)-3\) they indeed depend on the characteristic [Nagoya Math. J. 118, 203-216 (1990; Zbl 0707.13005)]. resolutions of determinantal ideals; Betti numbers; characteristic free representation Hashimoto, M.; Kurano, K.: Resolutions of determinantal ideals. Adv. math. 94, 1-66 (1992) Linkage, complete intersections and determinantal ideals, Determinantal varieties, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Resolutions of determinantal ideals: \(n\)-minors of \((n+2)\)-square matrices

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The paper deals with the mirror symmetry statement that, on the one hand, the generating function of intersection numbers on moduli spaces of curves in some variety (``A-side'') and, on the other, the function arising from the variation of Hodge structures on the ``B-side'' coincide. While, in the original version of \textit{V. V. Batyrev} and \textit{E. N. Materov} [Mosc. Math. J. 2, No. 3, 435--475 (2002; Zbl 1026.14016)], the ``Toric Residue Mirror Conjecture'' deals with the mutually dual families of CY-varieties arising from a pair of mutually dual reflexive polyhedra, the present paper proves this conjecture in a more general context -- reflexivity is not assumed anymore. Let \(\alpha:\mathbb Z^n\to M\cong\mathbb Z^r\), and denote by \(\beta:(\mathbb Z^n)^\ast\to{\widetilde N}\cong\mathbb Z^d\) the ``Gale-dual'', i.e.\ the dual map of \([{\widetilde M}:=\ker\alpha\hookrightarrow\mathbb Z^n]\). Under some technical conditions (\(\alpha\) is ``projective'' and ``spanning''), every \(u\in \alpha(\mathbb Q^n_{\geq 0})\) gives rise to a compact polytope \(\Delta_u:=\alpha^{-1}(u)\cap \mathbb Q^n_{\geq 0}\) sitting in a translate of \({\widetilde M}_\mathbb Q\). This translates into a \(d\)-dimensional projective toric variety \(V_A(c)\) (the ``A-side'') that depends only on the chamber \(c\) containing \(u\). The participating torus is \({\widetilde N}\otimes_\mathbb Z\mathbb C^\ast\), and the vector space \(M_\mathbb Q\) may be identified with \(H^2(V_A(c),\mathbb Q)\). In particular, polynomials \(P(z_1,\ldots,z_n)\) give rise to cohomology classes \(P(\alpha_1,\ldots,\alpha_n)\in S^{{\scriptscriptstyle \bullet}} M_\mathbb Q= H^{2{\scriptscriptstyle \bullet}}(V_A(c),\mathbb Q)\), and one may consider \(\int_{V_A(c)} P(\alpha_1,\ldots,\alpha_n)\). Using a result of \textit{M. Brion} and \textit{M. Vergne} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 715--741 (1999; Zbl 0945.32003)], this integral is representable as the so-called Jeffrey-Kirwan residue of \(P(\alpha)/\prod_i\alpha_i\), which in turn equals the integral of \(P(\alpha)/\prod_i\alpha_i\), now understood as a polynomial function on \(N_\mathbb R\), over some real \(r\)-cycle. Fixing a class \(\lambda\in c^\vee\subseteq N_\mathbb Q=H_2(V_A(c),\mathbb Q)\), one defines the so-called Morrison-Plesser moduli space \(\text{MP}_\lambda\) of \(\lambda\)-curves in \(V_A(c)\) in a similar manner as \(V_A(c)\) itself. For a polynomial \(P(z_1,\ldots,z_n)\), one defines \(\langle P\rangle_{\lambda,A,c}\) as an integral similar to that over \(V_A(c)\) -- yielding the generating function \(\langle P\rangle_{A,c}:=\sum_\lambda \langle P\rangle_{\lambda,A,c}\cdot z^\lambda\). It can be presented as the integral of a real form \(\Lambda\) over the \(r\)-cycle on \(N_\mathbb R\) mentioned above. The ``B-side'' is the \(d\)-dimensional, polarized toric variety \(V_B\) defined by the convex hull of the points \(\beta_1,\ldots,\beta_n\). Now, every polynomial \(P(z_1,\ldots,z_n)\) gives rise to a global section \(P(z_1\beta_1,\ldots,z_n\beta_n)\) of a tensor power of the ample line bundle, i.e.\ to an element of the homogeneous coordinate ring. Applying Cox's toric residue (also depending on \(z\) [ cf. \textit{D. Cox}, Ark. Mat. 34, 73--96 (1996; Zbl 0904.14029)]), one defines \(\langle P\rangle_B(z):= \text{TorRes}_z P(z_1\beta_1,\ldots,z_n\beta_n)\). This function may be obtained via cumulating values of a certain function on a finite subset of the embedded torus \({\widetilde M}\otimes_\mathbb Z\mathbb C^\ast\). Now, the miracle is that this finite set can naturally be seen as a subset of \(N_\mathbb C\) -- hence, applying the functions on it involves restricting them to \(N_\mathbb C\), i.e.\ applying \(\alpha\). Writing the result as an integral via using ordinary residues, one is surprised once more: The integral involves the same \(r\)-form \(\Lambda\) as we met on the A-side. Eventually, comparing several real \(r\)-cycles in \(N_\mathbb C\), the conjecture reduces to a new residue formula for the intersection pairings of toric manifolds. The method of the proof uses tropical geometry. The authors reach the result that \(\langle P\rangle_{A,c}(z)=\langle P\rangle_B(z)\). The right hand side does not depend on \(c\). However, on the left hand side, different \(c\)'s lead to different regions of convergence. toric varieties A. Szenes and M. Vergne, \textit{Toric reduction and a conjecture of Batyrev and Materov}, \textit{Invent. Math.}\textbf{158} (2004) 453 [math/0306311]. Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Toric reduction and a conjecture of Batyrev and Materov

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Over the complex numbers, the generic vanishing theorems on the cohomology groups \(h^i(X, \omega_X \otimes L)\) where \(L\) is an algebraically trivial line bundle on a smooth projective variety \(X\) with canonical bundle \(\omega_X\) are a fundamental tool in the study of varieties with maximal Albanese dimension. The proof of generic vanishing relies crucially on the results of \textit{J. Kollár} [Ann. Math. (2) 123, 11--42 (1986; Zbl 0598.14015); Ann. Math. (2) 124, 171--202 (1986; Zbl 0605.14014)] on the higher direct images \(R^ia_*\omega_X\) of the Albanese morphism \(a\colon X \to A\), which are known to fail in positive characteristic. In the previous article of \textit{C. D. Hacon} et al. [Duke Math. J. 168, No. 9, 1723--1736 (2019; Zbl 1436.14033)], the structure of Cartier module on \(R^ia_*\omega_X\) is exploited to obtain some generic vanishing statement, which are powerful enough to prove various birational characterisations of (ordinary) abelian varieties. In this paper, the authors further investigate generic vanishing theorems for Cartier modules on an abelian variety \(A\). The main result states that for a Cartier module \(F_*\Omega_0 \to \Omega_0\) on \(A\) and \(i >0\), then there exists a closed subset \(W_i\) of codimension \(i\) outside of which a Frobenius limit version of vanishing on \(h^i(A, \Omega_0 \otimes L)\) holds. Stronger results saying that \(W_i\) is a torsion translate of abelian subvarieties are proven in the case where \(A\) has no supersingular factors. One important application of this concerns the study of the singularities of theta divisors: an irreducible theta divisor of a principally polarised abelian variety without supersingular factors is strongly \(F\)-regular (generalising the fact that it has canonical singularities in characteristic 0). generic vanishing; theta divisors; positive characteristic Arithmetic ground fields for abelian varieties, Theta functions and abelian varieties, Vanishing theorems in algebraic geometry Generic vanishing in characteristic \(p>0\) and the geometry of theta divisors

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let H be an ample Cartier divisor on a normal projective surface Y over \({\mathbb{C}}\). Let R be the graded ring \(\oplus H^ 0(Y,{\mathcal O}(m(K_ Y+H)))\) for \(m\geq 0\). Define \(\kappa =tr.\deg.R-1\) \((\kappa =-\infty\) if \(R\cong {\mathbb{C}})\). The main result is the classification of the pair (Y,H) in terms of \(\kappa\). For the smooth case this was done by \textit{A. J. Sommese} [Duke Math. J. 46, 377-401 (1979; Zbl 0415.14019)] and by \textit{A. Lanteri} and \textit{M. Palleschi} [J. Reine Angew. Math. 352, 15-23 (1984; Zbl 0535.14003)]. For the normal Gorenstein case this is due to Sommese. The proof uses the normal surface version of the Mori theory discussed by the author in Duke Math. J. 52, 627-648 (1985). The classification is as follows: \(\kappa =-\infty\) if and only if (Y,H) is among the following: \((i) ({\mathbb{P}}^ 2,{\mathcal O}(1)),\quad ({\mathbb{P}}^ 2,{\mathcal O}(2)),\quad ({\mathbb{Y}}_ e,H_ e)\) for \(e\geq 2\) where the \({\mathbb{Y}}_ e\) is obtained by contracting the base section of the rational ruled surface \({\mathbb{F}}_ e=:{\mathbb{P}}({\mathcal O}_{P^ 1}\oplus {\mathcal O}_{P_ 1}(-e)\), \(e\geq 2\), (ii) \({\mathbb{P}}^ 1\)-bundle over a smooth curve and \(Hf=1\) for a fibre f (scroll). - \(\kappa =0\Leftrightarrow K_ Y+H\sim 0.\quad \kappa =1\Leftrightarrow (Y,H)\) is a conic bundle with \(\kappa =1\). In case \(\kappa =2\), R is finitely generated if and only if Y is \({\mathbb{Q}}\)-Gorenstein. Furthermore, it is shown that \(\kappa\geq 0\) if and only if \(\dim H^ 0(Y,{\mathcal O}(K_ Y+H))>0.\) Kodaira dimension; ample Cartier divisor; normal surface [Sa] Sakai, F., Ample Cartier divisors on normal surfaces, J. reine und angew. Math.366 (1986), 121--128. Families, moduli, classification: algebraic theory, Special surfaces, Divisors, linear systems, invertible sheaves Ample Cartier divisors on normal surfaces

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The paper has two parts. In the first, one studies for non-isolated hypersurface singularities questions as: finite determinacy, unfoldings and deformations, the topology of the nearby fibres. Best results are given when the singular locus has dimension one. The key point is to fix an analytic germ \(\Sigma\) in \(({\mathbb{C}}^ m,0)\) and to look to holomorphic functions which contain \(\Sigma\) in their singular locus and to coordinate transformations which leave \(\Sigma\) invariant. This leads to the study of the corresponding right-equivalence relation and of two algebraic notions: the primitive ideal associated to \(\Sigma\) and Jacobi modules. The primitive ideal of \(\Sigma\) is given exactly by the functions as above, i.e. it is \(\{f| (f)+J_ f\subset I_{\Sigma}\}\) \((J_ f\) is the Jacobi ideal of f). A Jacobi module is nothing else than a quotient \(I_{\Sigma}/J_ f.\) In the second part (pure algebraic) one studies the depth and the projective dimension of the quotient of two ideals. This topic emerges naturally from the first part and is used in the first part. depth of a module; resolutions of Jacobi modules; non-isolated hypersurface singularities; unfoldings; deformations; Jacobi ideal; projective dimension Pellikaan, G. R.: Hypersurfaces singularities and resolutions of Jacobi modules. (1985) Local complex singularities, Complex singularities, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry Hypersurface singularities and resolutions of Jacobi modules

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(K\) be a field and \(I\subset K[x_1, \ldots, x_n]\) an ideal. For \(w\in (\mathbb{R}_{\geq 0})^n\) let In\(_w(I)\) be the weighted tangent cone (defined by the weight vector \(w\)). The definitions of Ehlers, Khovanskii and Kouchnirenko of Newton non--degeneracy for hypersurfaces respectively complete intersections is generalized to arbitrary ideals as follows: Assume \(0\in V(I)\subseteq K^n\) is a singular point. The singularity \(0\in V(I)\) is Newton non--degenerate if for every \(w\in (\mathbb{R}_{\geq 0})^n\) the variety defined by In\(_w(I)\) does not have singularities in \((K^\ast)^n\). It is proved that Newton non--degenerate singularities can be resolved by a toric modification. The toric modification can be described in terms of the Gröbner fan of \(I\). Newton non-degenerate; Tropical Geometry; Singularity Aroca, F.; Gomez-Morales, M.; Shabbir, K., Torical modification of Newton non-degenerate ideals, \textit{Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM}, 107, 1, 221-239, (2013) Singularities of surfaces or higher-dimensional varieties, Toric varieties, Newton polyhedra, Okounkov bodies Torical modification of Newton non-degenerate ideals

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. -- We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author. toric variety; logarithmic double complex; de Rham complex; simplicial fan; hypercohomology groups; vanishing theorems; Ishida's cohomology groups T. Oda: ''The algebraic de Rham theorem for toric varieties'',Tohoku Math. J., Vol. 2, (1993), pp. 231--247. Vanishing theorems in algebraic geometry, de Rham cohomology and algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies The algebraic de Rham theorem for toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We study the depth properties of certain direct image sheaves on normal varieties. Let \(f: Y\rightarrow X\) be a proper morphism of relative dimension \(d\) from a smooth variety onto a normal variety such that the preimage \(E\) of the singular locus of \(X\) is a divisor. We show that for any integer \(m>0\), the higher direct image \(R^df_*\omega^{\otimes m}_Y(aE)\) modulo the torsion subsheaf is \(S_2\), provided that \(a\) is sufficiently large. In case \(f\) is birational, we give criteria on \(a\) for the direct image \(f_*\omega_Y(aE)\) to coincide with \(\omega_X\). We also introduce an index measuring the singularities of normal varieties. direct images of twisted pluricanonical sheaves; normal varieties; reflexive sheaves; index of singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Sheaves in algebraic geometry On direct images of twisted pluricanonical sheaves on normal varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be an \(n\)-dimensional, projective toric variety with an ample line bundle \(\mathcal L\). It is well-known that, for \(k\geq n-1\), the sheaf \(\mathcal L^{\otimes k}\) is very ample and defines a projectively normal embedding. Moreover, \textit{K.~Nakagawa} and \textit{S. Ogata} [Manuscr. Math. 108, 33--42 (2002; Zbl 0997.14014)] showed that the defining ideal is generated by quadrics if \(k\geq n\). In the present paper, the author lowers the latter bound to \(k\geq n-1\), provided that the toric variety \(X\) is at least 3-dimensional. This assumption is indeed necessary, since, in case of a surface, \textit{R.J. Koelman}'s papers [Beitr. Algebra Geom. 34, No. 1, 57--62 (1993; Zbl 0781.14025); Tohoku Math. J., II. Ser. 45, No. 3, 385--392 (1993; Zbl 0809.14042)] show that the case \(k=n-1\) \((=1)\) fails. The appearence of the assumption ``\(\dim X\geq 3\)'' becomes quite natural when looking at the second result of Ogata. It states that if the surjectivity of \(\Gamma(\mathcal L)\otimes\Gamma(\mathcal L^k)\to\Gamma(\mathcal L^{k+1})\) is granted from a certain degree, then this implies the quadric property of the corresponding tensor power of \(\mathcal L\) if this power exeeds \(n/2\). The main technical tool for studying these questions is to consider the kernel of a map of type \(\Gamma(\mathcal L_1)\otimes\Gamma(\mathcal L_2)\otimes\Gamma(\mathcal L_3)\to \Gamma(\mathcal L_1\otimes\mathcal L_2\otimes\mathcal L_3)\). Toric varieties, Newton polyhedra, Okounkov bodies, Families, moduli, classification: algebraic theory, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) On quadratic generation of ideals defining projective toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(K\) be a field and \(X=(x_{ij})\) a \((m\times n)\)-matrix of indeterminates over \(K\), \(n\geq m\). With \(S=K[x_{ij}]\), \(X\) determines the generic \(S\)-linear map \(\phi:S^n\rightarrow S^m.\) Let \(\text{Spec}R\) be the locus in \(\text{Spec}S\) where \(\phi\) has non-maximal rank: \(R\) is the quotient of \(S\) given by the maximal minors of \(X\), and is the generic determinantal variety. The classical \(R\)-modules \(M_a=\text{cok}\bigwedge^a_S\phi\) are maximal Cohen-Macaulay and are resolved by the Buchsbaum-Rim complex. In this article the authors prove that the \((M_a)_a\) yields a kind of non-commutative desingularization of the singular variety \(\text{Spec} R\): For \(1\leq a\leq m\) put \(M_a=\text{cok}\bigwedge^a_S\phi\) and \(M=\bigoplus_a M_a\). Then \(E=\text{End}_R(M)\) is maximal Cohen-Macaulay as an \(R\)-module with finite global dimension. That is, \(E\) is a non-commutative desingularization of \(\text{Spec} R\). If \(m=n\) then \(R\) is the hypersurface \(R=S/(\det\phi)\) and so \(R\) is Gorenstein and the non-commutative desingularization is an example of a non-commutative crepant resolution. The authors give a description by generators and relations of the non-commutative resolution \(E\) by stating that \(E\) as a \(K\)-algebra is isomorphic to the path algebra \(K\tilde Q\) of some quiver \(\tilde Q\). The results above are purely algebraic, but are proved by relating them to algebraic geometry. The classical fact that \(\text{Spec} R\) has a Springer type resolution of singularities is frequently used: Define the incidence variety \[ \mathcal Z=\{([\lambda],\theta)\in\mathbb P^{m-1}(K)\times M_{m\times n}(K)|\lambda\theta=0\} \] with projections \(p^\prime:\mathcal Z\rightarrow\mathbb P^{m-1}\) and \(q^\prime:\mathbb Z\rightarrow\text{Spec} R\). The key geometric facts then include: The scheme \(\mathcal Z\) is projective over \(\text{Spec} R\), which is of finite type over \(K\). The \(\mathcal O_{\mathcal Z}\)-module \[ \mathcal T := p^{\prime\ast}\left(\bigoplus^m_{a=1}\left(\bigwedge^{a-1}\Omega_{\mathbb P^{m-1}}\right)(a)\right) \] is a classical tilting bundle on \(\mathcal Z\) , i.e. (1) \(\mathcal T\) is a locally free sheaf, in particular, a perfect complex on \(\mathcal Z\), (2) \(\mathcal T\) generates the derived category \(\mathcal D(\text{Qch}(\mathcal Z))\); \(\text{Ext}^\bullet_{\mathcal O_{\mathcal Z}}(\mathcal T, C)=0\) for a complex \(C\) in \(\mathcal D(\text{Qch}(\mathcal Z))\) implies \(C\cong 0\), (3) \(\text{Hom}_{\mathcal O_{\mathcal Z}}(\mathcal T,\mathcal T[i])=0\) for \(i\neq 0\), (4) \(M\cong \mathbf{R}q^\prime_\ast\mathcal T\) and (5) \(E\cong\text{End}_{\mathcal Z}(\mathcal T)\). These geometric considerations leads to an interesting and important result stating that the variety \(\mathcal Z\) is the fine moduli space for the \(\tilde Q\)-representations \(W\) of dimension vector \((1,m-1,\left(\begin{smallmatrix} m-1\\2\end{smallmatrix}\right),\dots,1)\) that are generated by the last component \(W_m\). The proofs of the results depends mostly on the explicit computation of the cohomology of certain homogeneous bundles on \(\mathcal P^{m-1}\), determination of higher direct images of twisted bundles of homomorphisms between the modules of differential forms and other technical results. The article is more or less self contained, containing e.g. the construction of the projective tautological Koszul complex. In addition, of interest in itself is a construction of projective resolutions from sparse spectral sequences. This is then used in to construct the non-commutative desingularization \(E\) above, with algebra structure given by the quiverized Clifford algebra and its presentation. Particularly nice is the treatment of the noncommutative desingularization as a moduli space for representations. It is really interesting to notice that the points in \(\mathcal Z\) corresponding to the simple representations in \(W\) as those lying over the non-singular locus of \(\text{Spec} R\). The article is strongly recommended to anyone who will understand this level of representation theory in the algebraic geometric view. Be prepared to use a lot of effort to go through all proofs in detail. non-commutative desingularization; sparse spectral sequence; simple representations; tautological Koszul complex; maximal Cohen-Macaulay; quivers; Clifford algebra Abhyankar, S.: Uniformization in a \( p\)-cyclic extension of a two dimensional regular local domain of residue field characteristic \( p\) . Festschrift zur Gedächtnisfeier für Karl Weierstrass 1815 - 1965, Wissenschaftliche Abhandlungen des Landes Nordrhein-Westfalen \textbf{33} (1966), 243-317, Westdeutscher Verlag, Köln und Opladen Noncommutative algebraic geometry, Cohen-Macaulay modules, Global theory and resolution of singularities (algebro-geometric aspects), Riemann-Roch theorems, Rings arising from noncommutative algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Representations of quivers and partially ordered sets Non-commutative desingularization of determinantal varieties. I

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The authors study the BPS invariants of the preferred Calabi-Yau resolution of ADE polyhedral singularities \(\mathbb{C}^{3}/G\) given by Nakamura's \(G\)-Hilbert schemes. Among other things, the authors determine the \(\mathbb{C}^{*}\) -fixed locus of the moduli space of BPS invariants. BPS invariants; polyhedral singularities; Calabi-Yau; G-Hilbert schemes Global theory and resolution of singularities (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) BPS invariants for resolutions of polyhedral singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Toric varieties provide a wealth of examples for performing explicit calculations in algebraic geometry. Since they can be described by a combinatorial construction, many of their invariants can be computed by purely combinatorial considerations. For example, the degree of a toric variety with respect to an equivariant ample line bundle can be computed as the volume of a certain associated polytope. Recently, \textit{P. Philippon} and \textit{M. Sombra} [J. Inst. Math. Jussieu 7(2), 327--373 (2008; Zbl 1147.11033)] extended this yoga to the arithmetic setting in order to give a very concrete description of the canonical height of a toric variety. In the first part of the article under review, the authors give a formula for the height of a projective toric variety \(Y_{Q, \alpha}\) over a number field with respect to an equivariant semi-positive adelic line bundle \(\overline{L}_{\Psi}\). (They restrict to the case in which the metrics at all finite places are algebraic and positive.) Their formula is a sum of local integrals over the polytope \(\Delta_Q\) associated to the algebraic line bundle \(L_{\Psi}\), and the integrand is the Legendre-Fenchel dual of the equivariant local Green's function for \(\overline{L}_{\Psi}\). In the second part of the article, the authors explain the calculation of heights for several special classes of examples using the Fubini--Study metric. They give beautiful formulas for the height of a rational normal curve of degree-\(n\) in \(\mathbb P^n\) and for the height of a toric projective space bundle. This short article is remarkably clearly written, with proofs relegated to a later article. Few explicitly computable examples exist in the literature of arithmetic intersection theory, and the introduction of these elegant ones arising from toric geometry is exceedingly welcome. height; toric variety; Arakelov theory Arithmetic varieties and schemes; Arakelov theory; heights, Toric varieties, Newton polyhedra, Okounkov bodies, Heights Height of toric subschemes and Legendre-Fenchel duality

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a toric variety. Given an integer \(c\geq 2\), the associated endomorphism \(\theta_c\) of \(X\) induced by a multiplication by \(c\) on the lattice defining \(X\) is called a dilation. Any sequence \((c_1,c_2, \dots )\) yields the sequence of endomorphisms \(\theta_{c_i}\) of the \(K\)-theory \(K_*(X)\) and the homotopy \(K\)-theory \(KH_*(X).\) The Dilation theorem states that on toric varieties one has an isomorphism \[ \varinjlim_{\theta_c}K_*(X) \cong \varinjlim_{\theta_c}KH_*(X). \] For \(k\) of characteristic zero this was proved by \textit{J. Gubeladze} for affine toric varieties (cf. [Invent. Math. 160, No. 1, 173--216 (2005; Zbl 1075.14051)]) and the general case was proved by the authors in [Trans. Am. Math. Soc. 361, No. 6, 3325--3341 (2009; Zbl 1170.19001)]. Let \(X_{R}=X\times \mathrm{Spec}(R).\) The main result of the paper is the Dilation theorem for \(X_{R}.\) As a consequence the authors obtain a conjecture of Gubeladze concerning the monoid algebras \(k[A]\) when \(k\) is any regular ring. This states that if \(A\) is a cancellative, torsion-free commutative monoid with no non-trivial units then for every sequence \((c_1,c_2,\dots )\) of integers \(\geq 2\) and every regular ring \(k\) containing a field, there is an isomorphism \[ K_*(k) \cong\varinjlim_{\theta_c}K_*(k[A]). \] dilation map; homotopy K-theory; toric variety Cortiñas, G., Haesemeyer, C., Walker, M., Weibel, C.: The \textit{K}-theory of toric varieties in positive characteristic. J. Topol. 7, 247-286 (2014) Étale and other Grothendieck topologies and (co)homologies, \(K\)-theory of schemes, \(K\)-theory and homology; cyclic homology and cohomology The \(K\)-theory of toric varieties in positive characteristic

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties [For the entire collection see Zbl 0723.00022.] The author makes a brief historical note on desingularization: there is an interesting survey of the techniques of \textit{Zariski}, \textit{Abhyankar} and \textit{Hironaka} and a clear summary of the classical proofs of desingularization in dimension 1 and 2. At the end, there is an announcement of a resolution theorem of dimension 3 singularities of the equation: \(T^ p-f(x_ 1,x_ 2,x_ 3)\) where \(p\) is the characteristic. The author gives some hints about this proof: there are 60 different cases which are controlled with numerical characters built with Newton polyhedras. Unfortunately, the author did not quote the last results of \textit{E. Bierstone} and \textit{P. D. Milman} [cf. Effective methods in algebraic geometry, Proc. Symp., Castiglioncello 1990, Prog. Math. 94, 11-30 (1991; Zbl 0743.14012) and J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007), of the reviewer in Géométrie algébrique et applications, C. R. 2. Conf. int., La Rabida 1984, Vol. I: Géométrie et calcul algébrique, Trav. Cours 22, 1-21 (1987; Zbl 0621.14015)] and of \textit{O. Villamayor} [``Patching local uniformizations'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 25 (1992)]. desingularization; resolution; dimension 3 singularities; Newton polyhedras Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Development of contemporary mathematics On the resolution of singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We describe projective resolutions of \(d\) dimensional Cohen-Macaulay spaces \(X\) by means of a projection of \(X\) to a hypersurface in \(d+1\)-dimensional space. We show that for a certain class of projections, the resulting resolution is minimal. resolutions; Cohen-Macaulay spaces Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Germs of analytic sets, local parametrization Projective resolutions associated to projections

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The aim of this basic paper is to prove finiteness of Monsky-Washnitzer cohomology for non singular affine varieties over a finite field \( k \) of characteristic \( p\). The proof relies on Dwork's exponential modules [\textit{B. Dwork}, ``A deformation theory for singular hypersurfaces'' in: Algebr. Geom., Bombay Colloq. 1968, 85-92 (1969; Zbl 0218.14014)] and runs along the method Monsky used for non singular varieties over a field of characteristic zero [\textit{P. Monsky}, ``Finiteness of De Rham cohomology'', Am. J. Math. 94, 237-245 (1972; Zbl 0241.14010)]. By means of De Jong alterations, Berthelot gave an another proof of the finiteness theorem [\textit{P. Berthelot}, Invent. Math. 128, No. 2, 329-377 (1997; Zbl 0908.14005)] ; the method used here is much more explicit and then, unlike Berthelot's one, seems adaptable for a possible extension to non constant coefficients. The main idea is to reduce the general situation to the one dimensional case. Namely the general finiteness theorem is deduced from the existence of an index for a class of ordinary \( p\)-adic differential equations \( M_{f,n,m} \) where \( f \) is a polynomial in \( n \) variables whose coefficients are Teichmüller representatives (in \( {\mathbb C}_{p}\)) of elements of \( k \) and \( m \) is an integer prime to \( p \) and bigger than the total degree of \( f\). These differential equations are shown to have only two singularities: a regular one at infinity and an irregular one in zero. Moreover, because of their geometric origin, they are endowed with a Frobenius structure hence are known to have an index [\textit{G. Christol} and \textit{Z. Mebkhout}, ``Sur le théorème de l'indice des équations différentielles \( p\)-adiques''. III, Ann. Math. (to appear)]. Actually the \( M_{f,n,m} \) build up a large set of examples and counter-examples of \( p\)-adic differential equations. Each step in the reducing process needs auxiliary results. Most of them are interesting by themselves. So are the Bertini theorem in characteristic \( p\), the \( p\)-adic Gysin exact sequence for all codimensions, the ``magic bound'' (whose proof is a beautiful application of the Fourier transform) and the basic comparison theorem between algebraic and analytic cohomologies for exponential modules. Hence the various sides of the article justify more than enough the effort needed by its numerous notations and involved computations. Bertini theorem; Betti numbers; finiteness of Monsky-Washnitzer cohomology; \(p\)-adic differential equations; characteristic \(p\); \(p\)-adic Gysin exact sequence Mebkhout, Z., Sur le théorème de finitude de la cohomologie \textit{p}-adique d\(###\)une variété affine non singulière, Amer. J. Math., 119, 1027-1081, (1997) \(p\)-adic cohomology, crystalline cohomology, Vanishing theorems in algebraic geometry, \(p\)-adic differential equations On the finiteness theorem of the \(p\)-adic cohomology of a non-singular affine variety

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The resolution of singularities of algebraic varieties defined over a field of characteristic zero by a sequence of blow-ups is a famous result proved by \textit{H. Hironaka} [Ann. Math. (2) 79, 109-203, 205-326 (1964; Zbl 0122.38603)], which has a lot of applications. Probably, relatively few mathematicians read the 200 page proof. This can change now. The article is arranged in a similar way to a talk in a colloquium (25\% should be understood by everyone, 25\% is for people who are interested, the next 25\% is for the specialists and the rest only the speaker will understand) with one difference: that also the rest is understandable. It starts with an overview explaining the result and giving rough ideas of the proof. The author suggests that very busy people should only read this. The next chapter gives an introduction to the main problems (choice of the centre of the blow-up, equiconstant points, improvement of singularities under blow-up including many examples). This is for the next 25. The next chapter (constructions and proofs) gives the technical details. It contains also several examples and a section on problems in positive characteristic. In an appendix, necessary basic facts from commutative algebra and the theory of blow-ups used in the previous chapters are collected. It is advisable for everybody interested in resolution of singularities to read this article. Hironaka resolution of singularities; blowing up H.HAUSER,\textit{The Hironaka theorem on resolution of singularities (or: A proof we always wanted to} \textit{understand)}, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403.http://dx.doi.org/ 10.1090/S0273-0979-03-00982-0.MR1978567 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Here we study the existence of zero-dimensional curvilinear schemes \(Z \subset \mathbb{P}^{n},\, n=2,3\), with the expected minimal free resolution and with card \((Z_{red}\) very small. In the plane we prove the existence of arbitrary degree connected curvilinear zero-dimensional schemes with the expected minimal free resolution. zero-dimensional curvilinear subscheme; Cohen-Macaulay type; maximal rank; generator Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves, Parametrization (Chow and Hilbert schemes) Connected zero-dimensional subschemes of a projective space and their minimal free resolution

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We give a characterization of Gorenstein toric Fano varieties of dimension \(n\) with index \(n\) among toric varieties. As an application, we give a stronger version of Fujita's freeness conjecture and also give a simple proof of Fujita's very ampleness conjecture on Gorenstein toric varieties. Fano variety; toric variety S. Ogata and H.-L. Zhao, A characterization of Gorenstein toric Fano \(n\)-folds with index \(n\) and Fujita's conjecture, preprint (2014); arXiv:1404.6870. Toric varieties, Newton polyhedra, Okounkov bodies, Fano varieties, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) A characterization of Gorenstein toric Fano \(n\)-folds with index \(n\) and Fujita's conjecture

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X \subset {\mathbb{P}}^r\) be a non degenerate irreducible projective variety (over an algebraically closed field). If deg\((X)=\)codim\((X)+2\) then \(X\) is said to be a variety of almost minimal degree. It is known that these varieties are of one of these two types: (1) \(X \subset {\mathbb{P}}^r\) is linearly normal and \(X\) is normal; (2) \(X \subset {\mathbb{P}}^r\) is not linearly normal or \(X\) is not normal. Moreover if (1) holds then \(X \subset {\mathbb{P}}^r\) is a normal Del Pezzo variety and if (2) occurs then \(X \subset {\mathbb{P}}^r\) is the projection of a minimal degree variety \(\tilde X \subset {\mathbb{P}}^{r+1}\) of codimension greater than or equal to 2 from a closed point \(p \in {\mathbb{P}}^{r+1} \setminus \tilde X\). The paper under review is concerned with local aspects varieties of this type (2). First (see Section 3) it is shown that if \(X\) is not normal and \(x \in X\) is not a vertex then the dimension of the tangent to \(X\) at \(x\) is \(2\dim(X)+2-\mathrm{depth}(X)\), being depth\((X)\) the artihmetic depth of \(X\) (cf. Theorem 3.9). Moreover the multiplicity of \(x\) is \(2\). Second (see Section 5 and 6) a study of the embedding scrolls \(Y \subset {\mathbb{P}}^r\) is provided. These are defined for \(\deg(X) \geq 5\) as scrolls \(Y \subset {\mathbb{P}}^r\) containing \(X\) with \(\dim(Y)=\dim(X)+1\) (let us recall that their existence is known). It is shown that when \(X\) is not a cone singular embedding scrolls are always of the type \(\mathrm{Join}(\mathrm{Sing}(X),X)\) and either smooth embedding scrolls do not exist (\(2 \leq \mathrm{depth}(X) \leq \dim(X)\)) or are precisely described (\(\mathrm{depth}(X) = \dim(X)+1\)). When \(X\) is smooth all its embedding scrolls are smooth and the problem of their description is open. For part I, II, cf. [J. Pure Appl. Algebra 214, No. 11, 2033--2043 (2010; Zbl 1196.14047); Proc. Am. Math. Soc. 139, No. 6, 2025-2032 (2011; Zbl 1215.14052)]. almost minimal degree; tangent spaces; multiplicity; embedding scrolls M. Brodmann and E. Park, On varieties of almost minimal degree III: Tangent spaces and embedding scrolls. J. Pure Appl. Algebra 215 (2011), no. 12, 2859-2872 Varieties of low degree, Rational and unirational varieties On varieties of almost minimal degree. III: Tangent spaces and embedding scrolls

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(Y\) be an affine and smooth \(n\)-dimensional algebraic scheme of finite type over a field \(K\) and \(X\) a non-empty open subset of \(Y\). Here we prove that \(\text{cd}(X)\) is the first integer \(t\) such that \(h^i(X,{\mathcal O}_X)=0\) for all \(i>t\). Vanishing theorems in algebraic geometry Cohomological dimension of quasi-affine varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This is an announcement of the paper reviewed above. Euler characteristic; quotient singularity; crepant resolution; trihedral group Ito, Y.: Crepant resolution of trihedral singularities. Proc. Japan acad. Ser. A math. Sci. 70, 131-136 (1994) Global theory and resolution of singularities (algebro-geometric aspects), Homogeneous spaces and generalizations, Low codimension problems in algebraic geometry Crepant resolution of trihedral singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The families of affine semi-algebraic sets over a real-closed field K and semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properties of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation of an algebraic object weaker than a group and culminates with a version of the Jordan-Hölder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in \(K^n\) is \(n\)-dimensional over the scalar ultrafilters. affine real ultrafilters; affine semi-algebraic sets; Minkowki sums; cancellation; Jordan-Hölder theorem Real algebraic sets, Semialgebraic sets and related spaces An algebraic study of affine real ultrafilters

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Complex, quasi-smooth, projective toric varieties may be considered a generalization of the projective space \(\mathbb{P}^n\); they are obtained by glueing together affine pieces almost isomorphic to \(\mathbb{C}^n\). Those toric varieties may be given by a fan (a certain collection of rational, polyhedral cones in \(\mathbb{R}^n)\) containing all combinatorial information necessary for this process. In the present paper, the author describes a different method of synthesizing projective varieties as geometric quotients from the given fan. The relations to projective spaces are even more striking: First, assigning to each one-dimensional generator of the fan \(\Delta\) a coordinate, we obtain the affine space \(\mathbb{C}^{\Delta (1)}\) (in case of \(\mathbb{P} ^n\), this will be \(\mathbb{C}^{n + 1})\). Then, we have to do the following two jobs simultaneously: (i) Construct a certain subgroup of \((\mathbb{C}^*)^{\Delta(1)}\) by using the exact knowledge of the one-dimensional cones in \(\Delta\). This subgroup is isomorphic to some \((\mathbb{C}^*)^k\) and clearly acts on \(\mathbb{C}^{\Delta (1)}\). (In case of \(\mathbb{P}^n\), we have \(k = 1\).) (ii) Similarly to the construction of the Stanley-Reisner ring from a simplicial complex, the information which of the \(\Delta (1)\)-rays belongs to a common higher-dimensional cone (and which not) define a certain closed algebraic subset \(Z \subseteq \mathbb{C}^{\Delta (1)}\) of codimension at least two. \((Z \) equals \{0\} in case of \(\mathbb{P}^n\).) Now, the main result is that the toric variety assigned to a simplicial fan \(\Delta\) equals the geometric quotient \([\mathbb{C}^{ \Delta (1)} \backslash Z] / (\mathbb{C}^*)^k\). Since the group action is defined without using the information about incidences of \(\Delta\)-cones, this description is very useful for studying flips and flops, i.e. for changing \(\Delta\) without changing \(\Delta (1)\). Finally, this result is used for studying the automorphism group of a toric variety via considering \(\mathbb{C}^{\Delta (1)} \backslash Z\). In a paper of \textit{Daniel Bühler} (Diplomarbeit Zürich), this is generalized also to non-simplicial fans. geometric quotients of a fan; projective toric varieties; Stanley-Reisner ring; automorphism group Cox, D., The homogeneous coordinate ring of a toric variety, \textit{J. Algebra Geom.}, 4, 1, 17-50, (1995) Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients), Simplicial sets and complexes in algebraic topology, Birational automorphisms, Cremona group and generalizations, Homogeneous spaces and generalizations The homogeneous coordinate ring of a toric variety

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties It is well known that, while every projective surface can be projected isomorphically in \(\mathbb{P}^ 4\), in general the projection in \(\mathbb{P}^ 4\) of a smooth projective surface will have singularities. Therefore, the question of which surfaces admit an embedding in \(\mathbb{P}^ 4\) has been extensively studied. This paper is a nice survey of the topic. In the first section the author recalls general results and describes some examples; in the second section he focuses on the more specific question of which geometrically ruled surfaces can be embedded in \(\mathbb{P}^ 4\). embeddings of algebraic surfaces in \(\mathbb{P}^ 4\); ruled surfaces Embeddings in algebraic geometry, Projective techniques in algebraic geometry, Rational and ruled surfaces Embeddings of algebraic surfaces in \(\mathbb{P}^ 4\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This article presents a theory of equisingularity for families of \(0\)-dimensional sheaves of ideals on smooth algebraic surfaces in an arithmetic context. The value of this work is precisely this: It is pioneering in setting the formalisms in order that we may speak of equisingularity in the arithmetic case. It deals with families of \(0\)-dimensional schemes on regular surfaces, which is an interesting case to start with, since they appear both in the geometric theory of Enriques with the notion of proximity, and in the theory of Zariski of complete ideals in a \(2\)-dimensional regular local ring. As the authors say in the introduction, some of the results in the paper have been used by \textit{A. Nobile} and \textit{O. E. Villamayor} [Commun. Algebra 26, 2669-2688 (1998; Zbl 0938.14001)] to develop a theory analogous to Zariski's one for sheaves of ideals on an arithmetic \(3\)-fold. More precisely, the authors consider a Dedekind scheme \(T\) with the condition that, for all \(t \in T\), the residue field \(k(t)\) is a perfect field, and a smooth morphism \(\pi: X \rightarrow T\), where \(X\) is a \(3\)-dimensional scheme. By an arithmetic family of \(0\)-dimensional ideals on the fibers of \(X\) they mean a \(1\)-dimensional coherent sheaf of \({\mathcal O}_X\)-ideals \(I\) such that \(\pi\) induces a flat morphism \(V(I) \rightarrow T\), where \(V(I)\) is the subscheme defined by \(\sqrt I\). Then, at each point \(t \in T\), a geometric fiber of \(\pi\) at \(t\) is a smooth surface, and \(I\) induces a \(0\)-dimensional sheaf of ideals on it. The authors follow the notions appearing in a paper by \textit{J. J. Risler} [Bull. Soc. Math. Fr. 101, 3-16 (1973; Zbl 0256.14006)] in the context of local analytic geometry. The first condition they introduce, called (a) in the paper and generalizing (a) in the paper by Risler cited above, is the following: The morphism \(V(I) \rightarrow T\) is étale, if we consider the blowing up \(X_1 \rightarrow X\) at \(V(I)\) and the proper transform \(I_1\) of \(I\), then \(V(I_1) \rightarrow T\) is étale, and so on. The authors prove that this is equivalent to say that, for all geometric points \(\overline t\), the ideals \(I_{\overline t}\) have the same associated forests with the same weights given by the orders of the propers transforms (this is called condition (b)). A third approach is to add the proximity relations to these weighted forests (called then condition (c)) and to relate it to the equisingularity of \(T\)-curves \(C\) belonging to \(I\): To have equivalence the curves need to be ``general enough''. equisingularity; schemes over Dedekind rings; \(0\)-dimensional ideals; smooth surfaces; arithmetic \(3\)-fold [NV2]---, Arithmetic families of smooth surfaces and equisingularity of embedded schemes.Manuscripta Math., 100 (1999), 173--196. Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Dedekind, Prüfer, Krull and Mori rings and their generalizations Arithmetic families of smooth surfaces and equisingularity of embedded schemes

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The orbifold cohomology of complex orbifolds was introduced by \textit{W. Chen} and \textit{Y. Ruan} [Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)]. The article under review proposes a set of conjectures relating orbifold cohomology and the cohomology of certain resolutions. Supporting evidence for the conjectures is then offered in a number of interesting examples. In short, for those complex orbifolds admitting hyperkähler resolutions, Ruan conjectures that the cohomology of the resolution is isomorphic to the orbifold cohomology. This is termed ``the cohomological hyperkähler resolution conjecture''. It applies, for instance, to the case of the Hilbert scheme of points on a torus or a \(K3\) surface, when viewed as a hyperkähler resolution of the symmetric power of the surface. This particular case was in fact recently confirmed by \textit{M. Lehn} and \textit{C. Sorger} [Invent. Math. 152, No. 2, 305--329 (2003; Zbl 1035.14001)], \textit{B. Fantechi} and \textit{L. Göttsche} [Duke Math. J. 117, No. 2, 197--227 (2003; Zbl 1086.14046)] and \textit{B. Uribe} [Commun. Anal. Geom. 13, No. 1, 113--128 (2005; Zbl 1087.32012)]. A similar conjecture is formulated for general crepant resolutions of Gorenstein orbifolds. In this case one needs to include quantum corrections in the cohomology of the resolution. These quantum corrections encode certain Gromov-Witten invariants of exceptional fibers of the resolution, and vanish in the hyperkähler case. Finally, for \(K\)-equivalent manifolds, a cohomological miminmal model conjecture is formulated, also after including the quantum corrections in cohomology. Y. Ruan, ''The cohomology ring of crepant resolutions of orbifolds,'' in Gromov-Witten Theory of Spin Curves and Orbifolds, Providence, RI: Amer. Math. Soc., 2006, vol. 403. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds The cohomology ring of crepant resolutions of orbifolds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal \(\langle s, t \rangle \cap \langle u, v \rangle\) of the bigraded ring \(\mathbb{K}[s,t;u,v]\). Our analysis involves tools from algebraic geometry (Segre-Veronese varieties), classical commutative algebra (Buchsbaum-Eisenbud criteria for exactness, Hilbert-Burch theorem), and homological algebra (Koszul homology, spectral sequences). We treat in detail the case in which the bidegree is \((1, n)\). We connect our work to a conjecture of Fröberg-Lundqvist on bigraded Hilbert functions, and close with a number of open problems. bihomogeneous ideal; syzygy; free resolution; Segre map Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings defined by binomial ideals, toric rings, etc., Vanishing theorems in algebraic geometry The simplest minimal free resolutions in \({\mathbb{P}^1 \times \mathbb{P}^1}\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X=(X_{ij})_{1\leq i,j\leq n}\) be a symmetric matrix of indeterminates and let \(R=K[X]\) with K a field of characteristic zero. Let \(I_ p(X)\) be the determinantal ideal generated by all \(p\times p\) minors of X. It is known that \(I_ p(X)\) is a perfect ideal. Minimal free resolutions of \(R/I_ p(X)\) over R have been described by various authors in terms of Schur functors. Using the same methods the author constructs a minimal free resolution of \(R/I^ 2_{n-1}(X)\). The length of this resolution is 6, the depth of \(I^ 2_{n-1}(X)\) is 3 and it is no longer perfect. determinantal ideal; minimal free resolution Determinantal varieties, Complexes A resolution of the square of a determinantal ideal associated to a symmetric matrix

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author provides and discusses an analogue of a flasque resolution of a torus for a connected reductive linear algebraic group. Namely, by a flasque resolution of a group \(G\) one calls a central extension of algebraic groups \(1 \to S \to H\to G\to 1\), where \(H\) is a quasi-trivial group and \(S\) is a flasque torus. Here quasi-trivial means an extension of a quasi-trivial torus by a simply connected semisimple group. The paper is organized as follows: In sections 0-4 the author recalls basic properties of linear algebraic groups, establishes the existence and properties of such flasque and coflasque resolutions. In section 5 he discusses relations between resolutions and universal torsors of smooth compactifications of linear groups. In section 6 the author using the language of resolutions introduces and studies an algebraic fundamental group \(\pi_1(G)\) of a connected linear group \(G\). In section 7 he provides two formulas for the Brauer group of a smooth compactification of \(G\): one in terms of flasque resolutions of \(G\) and another in terms of the algebraic fundamental group of \(G\). In the last two sections he applies flasque resolutions to obtain information on \(R\)-equivalence classes of \(G(k)\). flasque resolution; linear algebraic group; universal torsor; algebraic fundamental group Colliot-Thélène, J.-L., Resolutions flasques des groupes linéaires connexes, J. Reine Angew. Math., 618, 77-133, (2008) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Galois cohomology of linear algebraic groups Flasque resolutions of linear connected groups.

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author constructs Galois coverings of the affine line over \(\mathbb{F}_ p\) with Galois group \(\text{SL}_ n (\mathbb{F}_{p^ r})\) (\(n\), \(r\) some positive integers) or a somewhat more general group \(G\). The idea is to take a connected algebraic group \(A\) over \(\mathbb{F}_ p\) and the Lang morphism \(L\) associated to an endomoprhism of \(A\), a power of which is a power of the Frobenius morphism; then the construction can be carried out with \(G= L^{-1}(1)\). Galois coverings of the affine line over finite field; Lang morphism Nori, M. V., \textit{unramified coverings of the affine line in positive characteristic}, Algebraic geometry and its applications (West Lafayette, IN, 1990), 209-212, (1994), Springer, New York Coverings of curves, fundamental group, Linear algebraic groups over finite fields, Finite ground fields in algebraic geometry Unramified coverings of the affine line in positive characteristic

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The classification of maximal Cohen-Macaulay (MCM) modules over Noetherian local rings is a difficult problem in general, and it has a long history; we refer the reader to the book [\textit{G. Leuschke} and \textit{R. Wiegand}, Cohen-Macaulay representations. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1252.13001)] for a detailed overview of the subject. Let \(k\) be an algebraically closed field such that \(\text{char}(k) \neq 2\). The goal of the article under review is to study the MCM representation type of rings of the form \[ k[[x,y,z]]/(xy, y^q - z^2), \] and also the ring \[ k[[x,y,z]]/(xy, z^2) \] (additional results in the case \(\text{char}(k) = 2\) are also obtained; see Remark 2.3 of the article under review for details). The main results of the article are as follows. The authors first prove that the above rings have \textit{tame} MCM representation type (see Section 4 of the article for the definition of tameness). They then go on to give an explicit description of all indecomposable MCM modules over the ring \(k[[x,y,z]]/(xy, z^2)\). Finally, the authors apply the previous results to construct explicit families of \textit{matrix factorizations} associated to the hypersurface ring \(k[[x,y]]/(x^2y^2)\). Matrix factorizations were introduced in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]; roughly speaking, they are the data of two-periodic tails of minimal free resolutions of finitely generated modules over hypersurface rings. maximal Cohen-Macaulay modules; matrix factorizations; MCM representation type Burban, I., Gnedin, W.: Cohen-Macaulay modules over some non-reduced curve singularities. arXiv:1301.3305v1 Cohen-Macaulay modules, Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry Cohen-Macaulay modules over some non-reduced curve singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The relation between Hodge theory of a smooth subcanonical \(n\)-dimensional projective variety \(X \subset \mathbb P_{\mathbb C}^N\) (i.e. \(\omega_X = \mathcal O_X(m)\) for some \(m\in \mathbb Z\)) and the deformation theory of the affine cone \(A_X\) over \(X\) is investigated. It is proved that there is a canonical isomorphism \[ (T_{A_X}^0)_m =H_{prim}^{n,0}(X) . \] If \(H^1(X,\mathcal O_X(k))=0\) for all \(k \in \mathbb Z\) then there is a canonical isomorphism \[ (T_{A_X}^1)_m =H_{prim}^{n-1,1}(X) . \] If also \(H^2(X,\mathcal O_X(k))=0\) for all \(k \in \mathbb Z\) then there is a canonical isomorphism \[ (T_{A_X}^2)_m =H_{prim}^{n-2,1}(X) . \] Moreover the whole primitive cohomology of \(X\) can be identified as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over \(X\). The results are used to compute Hodge numbers of smooth subcanonical projective varieties. The corresponding \textsc{Singular} code is given. Hodge theory; deformation theory; Hochschild cohomology; Singular Hodge theory and deformations of affine cones of subcanonical projective varieties Deformations of singularities, Transcendental methods, Hodge theory (algebro-geometric aspects), Deformations and infinitesimal methods in commutative ring theory, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Hodge theory and deformations of affine cones of subcanonical projective varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties This paper under review studies the geometry of affine Deligne-Lusztig varieties of hyperspecial level structure in both equi-characteristic and mixed characteristic settings. Let us introduce some notation. \par Let $F$ be a non-Archimedean local field, that is, $F$ is a finite extension of $\mathbb Q_p$ or of the field $\mathbb F_p((t))$ of Laurent series. Denote by $O_F$ and $k_F$ its ring of integers and its residue field, respectively, and fix a uniformizer $\epsilon$. Let $\Gamma$ be the absolute Galois group of $F$. Let $L$ denote the completion of the maximal unramified extension of $F$, $O_L$ its ring of integers, and $k$ the residue field. Denote by $\sigma$ the Frobenius map of $L/F$ and of $k/k_F$. \par Let $G$ be a reductive group scheme over $O_F$ and denote $K=G(O_L)$. Fix $S\subset T\subset B \subset G$, where $S$ is a maximal split torus, $T$ a maximal torus and $B$ a Borel subgroup. For each $\mu\in X_*(T)_{\text{dom}}$ and $b\in G(L)$, one associates the affine Deligne-Lusztig variety \[ X_\mu(b):=\{ gK\in G(L)/K \mid g^{-1} b \sigma(g)\in K \mu(\epsilon) K \}. \] This is a $k$-scheme locally of finite type when $\text{char\,} F=p>0$, and locally of perfectly finite type when $\text{char\,} F=0$ due to \textit{X. Zhu} [Ann. Math. (2) 185, No. 2, 403--492 (2017; Zbl 1390.14072)] and \textit{B. Bhatt} and \textit{P. Scholze} [Invent. Math. 209, No. 2, 329--423 (2017; Zbl 1397.14064)]. Thus, one can study the geometry of affine Deligne-Lusztig varieties even in the mixed characteristic case and non-minuscule coweights $\mu$. It is known that $X_\mu(b)$ is non-empty if and only if the $\sigma$-conjugacy class $[b]$ lies in the Kottwitz set $B(G,\mu)$. \par It was conjectured by Rapoport that $X_\mu(b)$ is equi-dimensional. At that time for the mixed characteristic case, the algebro-geometric structure of $X_\mu(b)$ was endowed by Rapoport-Zink spaces and the conjecture was first restricted to the EL or PEL-cases. In this paper the authors prove this conjecture in its great generality (Theorem 3.4). In particular, it holds true for all equi-characteristic cases and for the mixed characteristic cases where $\mu$ is minuscule, $G$ is classical, $p\neq 2$ and $F$ is unramified over $\mathbb Q_p$. This result is extended from results of \textit{U. Hartl} and \textit{E. Viehmann} [Adv. Math. 229, No. 1, 54--78 (2012; Zbl 1316.14089)], \textit{E. Viehmann} [J. Algebr. Geom. 17, No. 2, 341--374 (2008; Zbl 1144.14040); Doc. Math. 13, 825--852 (2008; Zbl 1162.14033)], \textit{P. Hamacher} [Int. Math. Res. Not. 2015, No. 23, 12804--12839 (2015; Zbl 1349.14156); Math. Z. 287, No. 3--4, 1255--1277 (2017; Zbl 1401.14193)], \textit{M. Chen} et al. [Compos. Math. 151, No. 9, 1697--1762 (2015; Zbl 1334.14017)] and Zhu [loc. cit.]. \par The authors' main result concerns the set $\Sigma(X_\mu(b))$ (resp.~$\Sigma^{\text{top}}(X_\mu(b))$) of (resp. top-dimensional) irreducible components of $X_\mu(b)$. It is known that the set $J_b(F)\backslash \Sigma(X_\mu(b))$ of orbits is finite, where $J_b$ is the twisted centralizer of $b$. For describing the results, we let $\widehat G$ be the dual group of $G$, which is equipped with a maximal split torus $\widehat S$ contained in a maximal $\widehat T$ such that there is a natural identification $X^*(\widehat T)=X_*(T)$. Consider an element $\lambda_G(b)\in X^*(\widehat T^\Gamma)$ which is a lift of the Kottwitz point $k_G(b)\in \pi_1(G)_\Gamma$ such that the restriction $\lambda=\lambda_G(b)|_{\widehat S}\in X^*(\widehat S)=X_*(S)$ is a ``best integral approximation'' of the Newton point $\nu_b\in X_*(S)_{\mathbb Q,\text{dom}}$. The main result (Theorem 5.12) of this paper describes the set $J_b(F)\backslash \Sigma^{\text{top}}(X_\mu(b))$ for minuscule $\mu$, which implies in particular the the following result. \par Theorem 1.4. Let $\mu\in X_*(T)_{\text{dom}}$ be minuscule, $b\in [b]\in B(G,\mu)$, and let $\tilde \lambda$ be a lift of $\lambda_G(b)$. Then there is a canonical surjective map \[ \phi: W\cdot \mu \cap \left (\tilde \lambda+(1-\sigma) X_*(T)\right )\to J_b(F)\ backslash \Sigma^{\text{top}}(X_\mu(b)). \] Moreover, this map is a bijection in the following cases: \par (1) $G$ is split and \par (2) the intersection $[b]\cap \text{Cent}_G(\nu_b)$ is a union of superbasic $\sigma$-conjugacy classes in $\text{Cent}_G(\nu_b)$. \par This gives a vast generalization of previous results of Viehmann [loc. cit.] for the cases $G=\mathrm{GL}_{n,\mathbb Q_p}$ or $G=\mathrm{GSp}_{2n,\mathbb Q_p}$ and of \textit{O. Bültel} and \textit{T. Wedhorn} [J. Inst. Math. Jussieu 5, No. 2, 229--261 (2006; Zbl 1124.11029)] and \textit{I. Vollaard} and \textit{T. Wedhorn} [Invent. Math. 184, No. 3, 591--627 (2011; Zbl 1227.14027)] for the case $G=\mathrm{GU}(1,n-1)_{\mathbb Q_p}$, where the set $J_b(F)\backslash\Sigma(X_\mu(b))$ is a singleton. The authors make a great contribution to the geometry of affine Deligne-Lusztig varieties. These results will be very useful for describing irreducible components of Newton strata of good reduction of more Shimura varieties. affine Deligne-Lusztig varieties; Rapoport-Zink spaces; affine Grassmannian Modular and Shimura varieties, Linear algebraic groups over local fields and their integers Irreducible components of minuscule affine Deligne-Lusztig varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the \(E_1\)-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic. Finally, we give a very powerful extension theorem for ample line bundles. O. Fujino, Fundamental theorems for the log minimal model program , to appear in Publ. Res. Inst. Math. Sci., preprint, Vanishing theorems in algebraic geometry, Classical real and complex (co)homology in algebraic geometry Vanishing theorems for toric polyhedra

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(k\) be a totally real cubic number field with ring of integers \({\mathcal O}_k\). The Hilbert modular threefold of \(k\) is a desingularization of the (natural) compactification of \(\text{PSL}_2 ({\mathcal O}_k)\setminus{\mathcal H}^3\). The goal of this paper is to prove that all rational Hilbert modular threefolds arise from fields with discriminant less than 75125. Specifically, it is shown that if \(k\) is a cubic field of discriminant at least 75125, then the arithmetic genus of the Hilbert modular variety of \(k\) is negative and hence the variety is not rational. Smaller bounds on the size of the discriminant are obtained for some special classes of cubic fields. Hilbert modular threefold; cubic field Grundman, H. G.: Nonrational Hilbert modular threefolds. J. number theory 83, 50-58 (2000) Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Modular and Shimura varieties, Zeta functions and \(L\)-functions of number fields Nonrational Hilbert modular threefolds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a nonsingular projective surface or a threefold which has a nef tangent bundle. In this paper the author proves that the motive of \(X\) is finite-dimensional in the sense of \textit{S.I. Kimura} [Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)], and gives an explicit Chow-Künneth decomposition. Moreover, he shows that both Murre's conjectures and the motivic Hard Lefschetz theorem hold for \(X\). He proves this result by showing firstly that if a nonsingular projective variety \(Y\) admits a finite cover \(Z\to Y\) such that \(Z\) is a relative cellular variety over an abelian variety, then \(Y\) has a Chow-Künneth decomposition. Armed with this result, he employs the classification theorem of surfaces and threefolds with a nef tangent bundle due to \textit{F. Campana} and \textit{T. Peternell} [Math. Ann. 289, No.1, 169--187 (1991; Zbl 0729.14032)], and completes the proof of the above result. Chow groups; motives; homogeneous spaces Iyer, J, Murre's conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle, Trans. Am. Math. Soc., 361, 1667-1681, (2008) Algebraic cycles, Structure of families (Picard-Lefschetz, monodromy, etc.), Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Murre's conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) denote a compact Riemann surface of genus \(g(X)> 0\) and \(L\) an ample line bundle on \(X\). \(L\) is said to be normally generated if, for each \(n>0\), the natural map \(\text{Sym}^n H^0(X, L)\to H^0(X, L^n)\) is surjective. Let \(\pi: X\to Y\) be a (possibly ramified) covering of compact Riemann surfaces and let \(g(Y)\) \((\geq 0)\) denote the genus of \(Y\). Problem: Classify ample line bundles on \(Y\) such that the pullbacks on \(X\) are normally generated. In this note, we will study this problem in the cases of \(\pi\) being double coverings with small \(g(X)\) or \(g(Y)\). Riemann surface; ample line bundle Global Riemannian geometry, including pinching, Special algebraic curves and curves of low genus, Vector bundles on curves and their moduli A note on normally generated line bundles on compact Riemann surfaces

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. automorphism group; affine cone; del Pezzo surface; infinite transitivity; flexibility; one-parameter unipotent group Perepechko, A.Yu.: Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5. Funct. Anal. Appl. \textbf{47}(4), 284-289 (2013) Automorphisms of surfaces and higher-dimensional varieties, Fano varieties, Group actions on affine varieties Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties As \textit{N. Spaltenstein} [Compos. Math. 65, 121--154 (1988; Zbl 0636.18006)] showed, the category of unbounded complexes of sheaves on a topological space has enough \(K\)-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives. unbounded complexes; Grothendieck category; resolution; triangulated category of equivariant motives; Zbl 0636.18006 C.~Serpé, \emph{Resolution of unbounded complexes in Grothendieck categories}, J. Pure Appl. Algebra \textbf{177} (200), no.~1, 103--112. DOI 10.1016/S0022-4049(02)00075-0; zbl 1033.18007; MR1948842 Grothendieck categories, Derived categories, triangulated categories, Chain complexes (category-theoretic aspects), dg categories, Étale and other Grothendieck topologies and (co)homologies Resolution of unbounded complexes in Grothendieck categories

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The paper under review consists of two parts. In the first part, the authors review some basic material about affine toric varieties and then they study more general ones. Their main concern are those toric varieties, not necessarily normal, that can be covered by a finite number of affine open sets, each one invariant under the action of the torus. For these, they provide a combinatorial description, in terms of fans where to each of their cones a finitely generated semigroup is attached, subject to suitable gluing conditions. Henceforth, ``toric variety'' will mean one in this sense. They discuss a number of properties of these varieties and other related concepts, such as invertible sheaves, ampleness, projectivity and blowing-ups with equivariant sheaves of ideals as centers. The blown-up variety is again toric. They emphasize the mentioned combinatorial description in their study. In the second part they study in detail certain blowing-ups, namely those whose center is the \textit{logarithmic Jacobian ideal} of the toric variety \(X\). The resulting variety is again toric, and if the base field has characteristic zero, this process agrees with the Semple-Nash modification of \(X\), where each point is replaced by the limit positions of tangent spaces at nearby regular points. This is more commonly called the Nash modification (or blow-up) of \(X\), but it seems that this process appeared for the first time in [\textit{J. G. Semple}, Proc. Lond. Math. Soc. (3) 4, 24--49 (1954; Zbl 0055.14505)]. They prove that, starting from a toric variety \(X\) and a monomial valuation \(V\) of maximal rank of its function field, dominating a point \(x\) of \(X\), successive blowing-ups centered at log Jacobian ideals uniformize the valuation. That is, we reach the situation of a toric variety \(X'\) and a regular point \(x' \in X'\), lying over \(x\), such that the valuation dominates \(x'\). The proof is complicated, and throughout they use the combinatorial description of the first part. In particular, in characteristic zero the uniformization is obtained by repeated application of the Semple-Nash modification. In the final sections of the article they re-interpret their uniformization result in terms of the Zariski-Riemann space of the fan associated to the toric variety \(X\), which involves the preorders of an underlying lattice, introduced in [\textit{G. Ewald} and \textit{M.-N. Ishida}, Tohoku Math. J. (2) 58, No. 2, 189--218 (2006; Zbl 1108.14039)]. In general the authors work over an arbitrary base field, and some results are still valid over more general rings. toric geometry; Semple-Nash modification; logarithmic Jacobian ideal; monomial valuation; uniformization González, P.D., Teissier, B.: Toric geometry and the Semple-Nash modification. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matemáticas 108(1), 1-48 (2014) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Modifications; resolution of singularities (complex-analytic aspects) Toric geometry and the Semple-Nash modification

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors. algebraic monoid; toric variety; solvable algebraic group; Demazure root; grading; locally nilpotent derivation Algebraic monoids, Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on affine varieties, Linear algebraic groups over arbitrary fields, Solvable groups, supersolvable groups Classification of noncommutative monoid structures on normal affine surfaces

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(T \simeq (\mathbb{C}^{*})^{n}\) be an \(n\)-dimensional torus and \(X^{*}(T)\) be its group of characters. A layer in \(T\) is a subvariety of \(T\) of the form \[ \mathcal{K}(\Gamma, \phi) := \{t \in T : \chi(t) = \phi(\chi) \text{ for all } \chi \in \Gamma\}, \] where \(\Gamma < X^{*}(T)\) is a split direct summand and \(\phi : \Gamma \rightarrow \mathbb{C}^{*}\) is a homomorphism. A toric arrangement \(\mathcal{A}\) is a finite set of layers \(\{\mathcal{K}_{1}, \dots, \mathcal{K}_{r}\}\) in \(T\). A wonderful model for the complement of an arrangement \(\mathcal{M}(\mathcal{A})\) is a smooth variety \(Y_{\mathcal{A}}\) containing \(\mathcal{M}(\mathcal{A})\) as an open set and such that \(Y_{\mathcal{A}} \setminus \mathcal{M}(\mathcal{A})\) is a divisor with normal crossings and smooth components. In this nicely written paper, the authors, in the first part, provide a short survey on the construction of a projective wonderful model \(Y_{\mathcal{A}}\) for a toric arrangement \(\mathcal{A}\) and the presentation of its cohomology ring by generators and relations. The second part of the paper explores the notion of well-connected building sets of subvarieties, which were introduced by the first two authors [Adv. Math. 327, 390--409 (2018; Zbl 1468.14090)], and it turned out that such sets play a crucial role in the computations of the cohomology rings of projective models for toric arrangements. toric arrangements; configuration spaces; compact models Configurations and arrangements of linear subspaces, Compactifications; symmetric and spherical varieties, Toric varieties, Newton polyhedra, Okounkov bodies On projective wonderful models for toric arrangements and their cohomology

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The author considers three-dimensional toric varieties given by a fan \(\Sigma\subseteq\mathbb{R}^3\). Under the assumption that its spherical section (i.e. \(\Sigma\cap S^2\)) is homeomorphic to a disk \(D^2\) he proves that the cohomology groups \(H^{kss\bullet}(X_\Sigma,\mathbb{Z})\) are free with the following dimensions: \(h^0=1\), \(h^2=\#\Sigma^{(1)}\), and \(h^4=\#(\int\Sigma)^{(1)}\). The assumption is particularly fulfilled for (partial) resolutions of three-dimensional affine toric varieties. This leads, as an application, to the verification of the integral MacKay correspondence for three-dimensional abelian quotient singularities. toric varieties; freeness of cohomology groups; McKay correspondence; three-dimensional abelian quotient singularities Étale and other Grothendieck topologies and (co)homologies, Toric varieties, Newton polyhedra, Okounkov bodies, \(3\)-folds, Singularities in algebraic geometry Integral cohomology of some smooth complex toric 3-folds

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Terminal singularities arise naturally in the theory of minimal models. Their analytic type is classified (works of Ried, Mori, Kollár, Shepard-Barron) in dimension 3. The paper under review studies the resolutions, some birational and combinatorial properties of 3-dimensional terminal singularities of type cD. It is shown that there is at most one non-rational exceptional divisor with discrepancy 1. Moreover, if such a divisor exists then it is birationally isomorphic to the product of the projective line with a hyperelliptic curve. terminal singularities Modifications; resolution of singularities (complex-analytic aspects), Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry On the resolution of 3-dimensional terminal singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a smooth affine complex threefold embedded into a smooth projective threefold \(T\) such that the reduced boundary divisor \(D\) has only simple normal crossings and the divisor \(K_T+D\) is not nef. The author is interested in the birational geometry of \(X\) and applies the \((K_T+D)\)-minimal model program \[ (T,D){\overset\varphi{^0}\dashrightarrow}(T^1,D^1)\ldots\dashrightarrow \dots{\overset\varphi^{s-1}\dashrightarrow}(T^s,D^s) \] to \((T,D)\). To keep the data of \(X\) under control in this process it is necessary to get a precise analysis of every step \(\varphi^i:(T^i,D^i)\dashrightarrow(T^{i+1},D^{i+1})\). The author gives an explicit description of the first step \(\phi^0\), namely a classification of the \((K_T+D)\)-negative extremal rays \(\mathbb R_+[C]\) on \(T\) and considers this result as basic for an inductive description of the following steps \(\phi^i\). The classification is carried out in a case-by-case investigation, based on the methods and results of the minimal model program. The main part of the classification is related to the situation \((K_T\cdot C)\geq 0\), when \(\log\) flips may occur. log canonical bundle; log flip; euclidean log flip S. Mori, Threefolds whose canonical bundles are not numerically effective , Minimal model program (Mori theory, extremal rays), Rational and birational maps, Classification of affine varieties Affine threefolds whose log canonical bundles are not numerically effective

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite sets of points in \(\mathbb{P}^1 \times \mathbb{P}^1\). Specifically, we describe a virtual resolution for a sufficiently general set of points \(X\) in \(\mathbb{P}^1 \times \mathbb{P}^1\) that only depends on \(| X |\). We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in \(\mathbb{P}^1 \times \mathbb{P}^1\); more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in \(\mathbb{P}^1 \times \mathbb{P}^1\). virtual resolutions; minimal graded resolutions; points; multi-projective spaces Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Toric varieties, Newton polyhedra, Okounkov bodies Virtual resolutions of points in \(\mathbb{P}^1 \times \mathbb{P}^1\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties For a prime power \(q>1\) and for \(n\in\mathbb N\), let \(A\) be the incidence matrix of points and hyperplanes in \(\mathbb P^n(\mathbb F_q)\); i.e., the square matrix of order \(\#\mathbb P^n(\mathbb F_q)\) whose rows and columns are in 1-1 correspondence with points and hyperplanes in \(\mathbb P^n\) rational over the finite field \(\mathbb F_q\), and whose entries are 1 or 0 depending on whether the point lies in the hyperplane or not, respectively. For a subset \(S\subseteq\mathbb P^n(\mathbb F_q)\) let \(A_S\) denote the corresponding \((\#S)\times(\#\mathbb P^n(\mathbb F_q))\) submatrix of \(A\). The \(p\)-rank of \(A_S\) was shown to be related to the Hilbert function of the homogeneous ideal \(I(S)\) of \(S\) by \textit{G. E. Moorhouse} [in: Mostly finite geometries. Proc. TED FEST '96 Conf., Iowa City 1996, Lect. Notes Pure Appl. Math. 190, 353-364 (1997; Zbl 0893.51012)]. Motivated by this, the present paper deduces some injectivity and surjectivity results for the restriction map \(H^0(X,\mathcal O_X(n))\to H^0(S,\mathcal O_X(n)|S)\) for certain smooth del Pezzo surfaces and for the Veronese surface. del Pezzo surface; Veronese surface; incidence matrix; Hilbert function Finite ground fields in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Rational points On the Hilbert function and the minimal free resolution of the \(\text{GF}(q)\)-points of del Pezzo surfaces of \(\mathbb{P}^n\)

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of \(p\)-divisible groups in positive characteristic and thus to arithmetic properties of Shimura varieties. We compare stratifications of affine Deligne-Lusztig varieties attached to a basic element \(b\). In particular, we show that the stratification defined by \textit{M. Chen} and \textit{E. Viehmann} [J. Algebr. Geom. 27, No. 2, 273--304 (2018; Zbl 1408.14141)] using the relative position to elements of the group \(\mathbb{J}_b,\) the \(\sigma \)-centralizer of \(b\), coincides with the Bruhat-Tits stratification in all cases of Coxeter type, as defined by \textit{X. He} and the author [Camb. J. Math. 3, No. 3, 323--353 (2015; Zbl 1350.14026)]. affine Deligne-Lusztig varieties Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Linear algebraic groups over local fields and their integers Stratifications of affine Deligne-Lusztig varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We show that many classical results of the minimal model program do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is not normal, a three dimensional klt singularity which is not rational nor Cohen-Macaulay, and a klt Fano threefold with non-trivial intermediate cohomology. Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry Purely log terminal threefolds with non-normal centres in characteristic two

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties A \textit{real form} of a complex algebraic variety \( V\) is an equivalence class of an anti-holomorphic involution \( \sigma:\mathbb{C}V\rightarrow \mathbb{C}V\) on the set of its complex points by the natural equivalence relation induced by automorphisms of the variety. The variety \( V \) identified with \(\mathbb{C}V\) is the \textit{complexification} of the real algebraic variety \( (V,\sigma)\). A classical problem in real algebraic geometry is the classification of real forms of a given complex algebraic variety. The first example of a complex projective variety with infinitely many non-isomorphic real forms was obtained by \textit{J. Lesieutre} [Invent. Math. 212, No. 1, 189--211 (2018; Zbl 1393.14012)] in dimension \(d\geq 6 \). It was then generalised by \textit{T.-C. Dinh} and \textit{K. Oguiso} [Duke Math. J. 168, No. 6, 941--966 (2019; Zbl 1427.14085)] to any dimension \(d\geq 2 \) for varieties of Kodaira dimension \( d - 2 \), and by \textit{T.-C. Dinh} et al. [``Projective rational manifolds with non-finitely generated discrete automorphism group and infinitely many real forms'', Preprint, \url{arXiv:2002.04737}] to any \(d\geq 3 \) for projective rational varieties. The question of existence of affine rational algebraic varieties with infinitely many real forms was open. The first main result of the paper under review fills this gap for smooth real rational affine fourfolds. For a real algebraic variety \( (V,\sigma)\), the set \(\mathbb{C}V\) viewed as a real set with the Euclidean topology is the fix-point set of the anti-holomorphic involution \( \Delta\circ(\sigma\times\sigma) \), where \( \Delta: \mathbb{C}V\times\mathbb{C}V\rightarrow\mathbb{C}V\times\mathbb{C}V\) is the reflection in diagonal. This construction is functorial in \( (V,\sigma)\). The authors apply it to the complex line bundles \(\mathcal{O}_{\mathbb{C}\mathbb{P}^1}(n)\rightarrow\mathbb{C}\mathbb{P}^1, n\geq 0\), over \(\mathbb{C}\mathbb{P}^1\simeq S^2\), which are pairwise non-homeomorphic since, for a given \( n \), the self-intersection of the zero section of the bundle is \( n \). In the paper, it is proved that the complexifications \(E_n\rightarrow\mathbb{S}^2\) of the bundles are all isomorphic to the trivial bundle \(\mathbb{S}^2\times\mathbb{C}^2\rightarrow\mathbb{S}^2\) where \( \mathbb{S}^2\subset \mathbb{C}^3\) is the complexification of \( S^2 \). Thus the authors obtain Theorem 1. The smooth rational affine fourfold \( \mathbb{S}^2\times\mathbb{C}^2 \) has at least countably infinitely many pairwise non-isomorphic real forms. Let \( X \) be a real rational affine variety of dimension \( m \) and of log-general type (i.e., smooth real rational affine variety whose complexification has logarithmic Kodaira dimension equal to \( m \)). E.g., let \( X \) be the complement in \( \mathbb{R}\mathbb{P}^n \) of a smooth real hypersurface of degree \( r > m+1 \) if \( m>1 \), and the complement of three general real points in \( \mathbb{R}\mathbb{P}^1\) if \( m=1 \). Taking the product of \( E_n \) with \( X \) allows the authors to derive the following result: Theorem 2. For every \( d\geq4 \), there exist smooth rational affine varieties of dimension \( d \) which have at least countably infinitely many pairwise non-isomorphic real forms. After the reviewed paper has appeared, \textit{T.-C. Dinh} et al. [``Smooth complex projective rational surfaces with infinitely many real forms'', Preprint, \url{arXiv:2106.05687}] described projective rational surfaces with infinitely many real forms, and \textit{A. Bot} [``A smooth complex rational affine surface with uncountably many real forms'', Preprint, \url{arXiv: 2105.08044}] constructed a smooth affine rational surface with the same property. rational varieties; affine varieties; real forms; real structures Topology of real algebraic varieties, Rational and unirational varieties, Rational and ruled surfaces, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Smooth rational affine varieties with infinitely many real forms

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields. Arithmetic ground fields for curves, Étale and other Grothendieck topologies and (co)homologies, Algebraic functions and function fields in algebraic geometry On residue maps for affine curves

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The following statement is a consequence of a theorem of Buchweitz and results on idempotent completions of triangulated categories: Let \(X\) be an algebraic variety with isolated Gorenstein singularities \(Z=\text{Sing}(X)=\{x_1,\dots,x_p\}\). Then there is an equivalence of triangulated categories \[ \left(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\right)^\omega\overset\sim\rightarrow\bigvee_{i=1}^p \underline{\text{MCM}}\left(\hat{\mathcal O}_{X,x_i}\right). \] The left hand side stands for the idempotent completion of the Verdier quotient \(\frac{D^b(\text{Coh}(X))}{\text{Perf}(X)}\), on the right-hand side \(\underline{\text{MCM}}(\hat{\mathcal O}_{X,x_i})\) denotes the stable category of maximal Cohen-Macaulay modules over \(\hat{\mathcal O}_{X,x_i}=:\hat O_i\). The main goal of this article is to generalize this construction as follows. Let \(\mathcal F^\prime\in\text{Coh}(X),\) \(\mathcal F:=\mathcal O\oplus\mathcal F^\prime\) and \(\mathcal A:=\mathcal End_X(\mathcal F)\). Consider the ringed space \(\mathbb X:=(X,\mathcal A)\). It is well known that the functor \(\mathcal F\overset{\mathbb L}\otimes_X-:\text{Perf}(X)\longrightarrow D^b(\text{Coh}(\mathbb X))\) is fully faithful. If \(\text{gl.dim}(\text{Coh}(\mathbb X))<\infty\) then \(\mathbb X\) can be viewed as a non-commutative (or categorical) resolution of singularities of \(X\), and the authors suggest to study the triangulated category \(\Delta_X(\mathbb X):=(\frac{D^b(\text{Coh}(\mathbb X))}{\text{Perf}(X)})^\omega\) called the relative singularity category. Assuming \(\mathcal F\) to be locally free on \(U=X\backslash Z\), an analogue of the ``localization equivalence'' is proved for the category \(\Delta(\mathbb X)\). In addition, the Grothendieck group \(\Delta(\mathbb X)\) is described. The main result of the article is a complete description of \(\Delta_U(\mathbb Y)\) in case \(Y\) is an arbitrary curve with nodal singularities and \(\mathcal F^\prime:=\mathcal I_Z\) is the ideal sheaf of the singular locus of \(Y\). It is proved that \(\Delta_Y\) splits into a union of \(p\) blocks: \(\Delta_Y(\mathbb Y)\overset\sim\rightarrow\bigvee_{i=1}^p\Delta_i\), where \(p\) is the number of singular points of \(Y\). Each block is equivalent to the category \(\Delta_{\text{nd}}=\frac{\text{Hot}^b(\text{pro}(A_{nd}))}{\text{Hot}^b(\text{add}(P_\ast))},\) where \(A_{nd}\) is the completed path algebra of the quiver with nodes \(-,\ast,+\) and arrows \(\alpha=(-\ast),\beta=(\ast-),\delta=(\ast+),\gamma=(+\ast)\) and relations \(\delta\alpha=0\), \(\beta\gamma=0\), and \(P_\ast\) is the indecomposable projective \(A_{nd}\)-module corresponding to the vertex \(\ast\). The authors prove that the category \(\Delta_{nd}\) is idempotent complete and \(\text{Hom}\)-finite, and moreover, they give a complete classification of indecomposable objects of \(\Delta_{nd}\). \(\Delta_{nd}\overset\sim\rightarrow(\frac{D^b(\Lambda-\text{mod})}{\text{Band}(\Lambda)})^\omega\) where \(\Lambda\) is the path algebra of the quiver with nodes \(1,2,3\), arrows \(a,c\) from \(1\) to \(2\) and \(b,d\) from \(2\) to \(3\) and relations \(ba=0\), \(dc=0\), and \(\text{Band}(\Lambda)\) is the category of the band objects in \(D^b(\Lambda-\text{mod})\), i.e. the objects which are invariant under the Auslander-Reiten translation in \(D^b(\Lambda-\text{mod})\). Thus the Auslander-Reiten quiver of \(\Delta_{\text{nd}}\) is described. Let \(\text{P}(X)\) be the essential image of \(\text{Perf}(X)\) under the fully faithful functor \(\mathbb F:=\mathcal F\overset{\mathbb L}\otimes-:\text{Perf}(X)\rightarrow D^b(\text{Coh}(\mathbb X))\). This category is described in the following intrinsic way: \(\text{Ob}(\text{P}(X))=\{\mathcal H^\bullet\in \text{Ob}(D^b(\text{Coh}(\mathbb X)))|\mathcal H^\bullet\in\text{Im}(\text{Hot}^b(\text{add}(\mathcal F_x))\rightarrow D^b(\mathcal A_X-\text{mod}))\}\). In the above notations, the relative singularity category \(\Delta_X(\mathbb X)\) is the idempotent completion of the Verdier quotient \(\text{D}^b(\text{Coh}(\mathbb X))/\text{P}(X)\), and it has a natural structure of triangulated category. Some of the main results are thus: Let \(D^b_Z(\text{Coh}(\mathbb X))\) be the full subcategory of \(D^b(\text{Coh}(\mathbb X))\) consisting of complexes whose cohomology is supported in \(Z\) and \(\text{P}_Z(X)\cap D^b_Z(\text{Coh}(\mathbb X))\). Then the canonical functor \(\mathbb H:\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)}\rightarrow\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)}\) is fully faithful. With the same notations, the authors prove that the induced functor \(\mathbb H^\omega:(\frac{D^b_Z(\text{Coh}(\mathbb X))}{\text{P}_Z(X)})^\omega\rightarrow(\frac{D^b(\text{Coh}(\mathbb X))}{\text{P}(X)})^\omega \) is an equivalence of triangulated categories. In addition to much more, the authors answer the questions: Is the category \(\Delta_Y(\mathbb Y)\) \(\text{Hom}\)-finite? What are the indecomposable objects? What is the Grothendieck group of \(\Delta_Y(\mathbb Y)\)? Assume that \(E\) is a plane nodal cubic curve. What is the relation of \(\Delta_E(\mathbb E)\) with the ``quiver description'' of \(D^b(\text{Coh}(\mathbb E))\)? To conclude in line with the authors: Let \(Y\) be a nodal algebraic curve, \(Z\) its singular locus, \(\mathcal I=\mathcal I_Z\) and \(\mathbb Y=(Y,\mathcal A)\) for \(\mathcal A=\mathcal End_Y(\mathcal O\oplus I)\). Similarly, let \(O=k[[u,v]]/(uv)\), \(\mathfrak m=(u,v)\) and \(A=\text{End}_O(O\oplus\mathfrak m)\). Then the following results are true: The category \(\Delta_Y(\mathbb Y)\) splits into a union of \(p\) blocks \(\Delta_{nd}\), where \(p\) is the number of singular points of \(Y\) and \(\Delta_{nd}=\Delta_O(A)\). The category \(\Delta_{nd}\) is \(\text{Hom}\)-finite and representation discrete. In particular its indecomposable objects and the morphism spaces between them are explicitly known. Moreover, one can compute its Auslander-Reiten quiver. It is proved that \(K_0(\Delta_{nd})\cong\mathbb Z^2\). The category \(\Delta_{nd}\) admits an alternative ''quiver description'' in terms of representations of a certain gentle algebra \(\Lambda\). A nontrivial article, not very self contained, but the ideas are easy to follow and very interesting. triangulated category; non-commutative resolution; K-theory; Auslander-Reiten quiver; Grothendieck group; Homotopy category Burban, I; Kalck, M, The relative singularity category of a non-commutative resolution of singularities, Adv. Math., 231, 414-435, (2012) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Derived categories, triangulated categories The relative singularity category of a non-commutative resolution of singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties If \(X\) is a toric variety and \(H\) is a subtorus of the big torus then \(X\) has a toric quotient, i.e. a categorical quotient \(p:X\to X/_{ \text{tq}}H\) in the category of toric varieties. This was shown by the authors of the present paper [\textit{A. A'Campo-Neuen} and \textit{J. Hausen}, Tohoku Math. J., II. Ser. 51, 1-12 (1999; Zbl 0942.14028)]. A divisorial variety as defined by Borelli is an irreducible variety in which every point has an affine neighbourhood whose complement supports an effective Cartier divisor: For toric varieties this is the same as the condition of \textit{T. Kajiwara} [Tohoku Math. J., II. Ser. 50, 139-157 (1998; Zbl 0949.14032)] of having enough invariant effective Cartier divisors. In this paper the authors construct a toric divisorial reduction \(Y\to Y^{ \text{tdr}}\) from an arbitrary toric variety \(Y\) to a divisorial toric variety \(Y^{ \text{tdr}}\), universal for maps from \(Y\) to toric divisorial varieties. They show that the action of \(H\) on a divisorial toric variety \(X\) admits a categorical quotient in the category of divisorial varieties if and only if \(q\circ p:X\to (X/_{ \text{tq}}H)^{ \text{tdr}}\) is surjective, where \(q:X/_{ \text{tq}}H\to (X/_{ \text{tq}}H)^{ \text{tdr}}\) is the divisorial reduction of \(X/_{ \text{tq}}H\). In that case \(q\circ p\) is the categorical quotient. It is an open problem to find necessary and sufficient conditions for a categorical quotient of a given toric variety by a subtorus in the category of algebraic varieties to exist. toric variety; categorical quotient; geometric invariant theory; divisorial variety; toric divisorial reduction A. A'Campo-Neuen, J. Hausen: Quotients of divisorial toric varieties. Michigan Math. J., Vol. 50, No. 1, 101--123 Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Geometric invariant theory Quotients of divisorial toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X_0\) be a smooth geometrically connected variety defined over a finite field \(\mathbb{F}_q\), and let \(\mathcal{E}_0^\dagger\) be an irreducible overconvergent \(F\)-isocrystal on \(X_0\). We show that if a subobject of minimal slope of the associated convergent \(F\)-isocrystal \(\mathcal{E}_0\) admits a non-zero morphism to \(\mathcal{O}_{X_0}\) as a convergent isocrystal, then \(\mathcal{E}_0^\dagger\) is isomorphic to \(\mathcal{O}_{X_0}^\dagger\) as an overconvergent isocrystal. This proves a special case of a conjecture of Kedlaya. The key ingredient in the proof is the study of the monodromy group of \(\mathcal{E}_0^\dagger\) and of the subgroup defined by \(\mathcal{E}_0\). The new input in this setting is that the subgroup contains a maximal torus of the entire monodromy group. This is a consequence of the existence of a Frobenius torus of maximal dimension. As an application, we prove a finiteness result for the torsion points of abelian varieties, which extends the previous theorem of Lang-Néron and answers positively a question of Esnault. isocrystals; slope filtration; abelian varieties \(p\)-adic cohomology, crystalline cohomology, Arithmetic ground fields for abelian varieties Maximal tori of monodromy groups of \(F\)-isocrystals and an application to abelian varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The authors investigate a procedure to resolve singularities that has reasonable ``functorial'' properties. The main result is expressed in terms of ``marked ideals'', that is a 5-tuple \(\mathcal I = (M,N,E,I,d)\) where \(N\) is a subvariety of \(M\), both smooth over a field of characteristic zero, \(I\) is a coherent sheaf of ideals of \(N\), \(E\) is a normal crossings divisor of \(M\), transversal to \(N\) and \(d\) is a positive integer. The singular locus, or cosupport, of \(\mathcal I\) is the set of points \(x\) of \(N\) such that the order of the stalk \(I_x\) is \(\geq d\). One may define the transformation of such a marked ideal with center a suitable smooth subvariety of \(N\) (the \textit{admissible} transformations), the result is a new marked ideal. The objective is to obtain, by means of a finite sequence of admissible transformations, a marked ideal whose cosupport is empty. Such a sequence is called a resolution sequence. If this can be done in a reasonable constructive (or algorithmic) way other more classical desingularization theorems follow rather easily. (This is explained in the present article). The authors define a notion of equivalence of marked ideals as follows. A test transformation is either an admissible one, or one determined by the blowing-up of a center which is the intersection of two components of \(E\), or one induced by a projection \(M \times {\mathbb{A}}^1 \to M\). Two marked ideals \(\mathcal I = (M,N,E,I,d)\) and \(\mathcal J = (M,N',E,J,d')\) are equivalent if they have the same sequences of of test transformations. Then they prove that there is way to associate to each marked ideal a resolution sequence such that if \(\mathcal I\) and \(\mathcal J\) are equivalent, then the resolution sequence associated to \(\mathcal I\) is obtained by using the same centers as were needed for the sequence of \(\mathcal J\). Moreover, this procedure is compatible with smooth morphisms \(M' \to M\). This is what is meant by functoriality of the process. The procedure is a variant of that introduced by the authors in their fundamental paper \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)] incorporating some ideas from \textit{J. Włodarczyck}'s article [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. The exploitation of the explicit requirements on functoriality simplifies the verification of the fact that certain constructions are independent of the elections made, which is a hard problem in this type of desingularization work. The authors use the functorial character of the algorithm to show that it coincides with others recently introduced by J. Wlodarczyk and J. Kollár. resolution; blowing-up; marked ideal; admissible transformation; derivative ideals; test transformation Bierstone, E., Milman, P.: Functoriality in resolution of singularities. Publ. R.I.M.S. Kyoto Univ.~\textbf{44}, 609-639 (2008) Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Birational geometry, Modifications; resolution of singularities (complex-analytic aspects), Equisingularity (topological and analytic) Functoriality in resolution of singularities

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The paper refers to the problems of algebraic geometry. The subject of study is the complex toric variety of \(X(\Delta)\) associated to a smooth fan \(\Delta\). The fundamental group of \(X(\Delta)\) and the universal cover of \(X(\Delta)\) are decribed in terms of the fan with reference to the motivation in the paper [\textit{M. W. Davis} and \textit{T. Januskiewicz}, Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)]. Necessary and sufficient conditions on \(\Delta\) are given under which \(X(\Delta),\pi_1(X(\Delta))\) are aspherical or abelian, respectively. Among the areas examined there are also conditions necessary and sufficient for the \(C_\Delta\) to be a \(K(\pi,1)\) space, whereas \(C_\Delta\) is the complement of a real subspace arrangement associated to \(\Delta\). V Uma, On the fundamental group of real toric varieties, Proc. Indian Acad. Sci. Math. Sci. 114 (2004) 15 Toric varieties, Newton polyhedra, Okounkov bodies, Topology of real algebraic varieties On the fundamental group of real toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\subset\mathbb{P}^r\) be a non-degenerate variety and \(\Gamma(X)\) the closure in \(X\) of the set of all points \(P \in X\) such that the projection of \(X\) from \(P\) is not birational. Here we study the irreducible components of \(\Gamma(X)\), using proofs and ideas contained in a paper by \textit{A. Calabri} and \textit{C. Ciliberto} [Adv. Geom. 1, No. 1, 97--106 (2001; Zbl 0981.14010)] concerning the outer non-birational projections. Ballico, E., Special inner projections of projective varieties, Ann. Univ. Ferrara Sez. VII (N.S.), 0430-3202, 50, 23-26, (2004) Projective techniques in algebraic geometry Special inner projections of projective varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties For a complex semisimple group \(G\) the associated affine Grassmannian is \(\mathcal{G}r=G((t^{-1}))/G[t].\) In this paper using \(G=\text{SL}_{n}\), the authors describe the ideal of transversal slices \(\mathcal{G}r^{\lambda}_{\mu}\) (\(\mu \leq \lambda\) is a pair of dominant coweights) to spherical Schubert varieties in the affine Grassmannian. \(\mathcal{G}r^{\lambda}_{\mu}\) is the vanishing of a Poisson ideal \(J^{\lambda}_{\mu}\). Letting \(\mathcal{X}^{\lambda}_{\mu}\) be the scheme whose ideal is \(J^{\lambda}_{\mu}\) the authors prove that for \(G=\text{SL}_{n}\) the scheme \(\mathcal{X}^{\lambda}_{\mu}\) is reduced. affine Grassmannian; Poisson structure; spherical Schubert variety; Weynman's equations. Loop groups and related constructions, group-theoretic treatment, Grassmannians, Schubert varieties, flag manifolds Reducedness of affine Grassmannian slices in type A

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(T\) be a torus over an algebraically closed field \(\mathbb K\) of characteristic 0, and consider a projective \(T\)-module \(\mathbb P(V)\). We determine when a projective toric subvariety \(X\subset \mathbb P(V)\) is self-dual, in terms of the configuration of weights of \(V\). Bourel, M; Dickenstein, A; Rittatore, A, Self-dual projective toric varieties, J. Lond. Math. Soc. (2), 84, 514-540, (2011) Toric varieties, Newton polyhedra, Okounkov bodies, Projective techniques in algebraic geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Group actions on varieties or schemes (quotients) Self-dual projective toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(A\) be an ample line bundle on a projective toric variety \(X\) of dimension \(n (\geq 2)\). It is known that the \(d\)-th tensor power \(A^{\otimes d}\) embeds \(X\) as a projectively normal variety in \(\mathbb{P}^r := \mathbb{P}(H^0(X,L\otimes d))\) if \(d\geq n - 1\). In this paper first we show that when \(\dim X = 2\) the line bundle \(A^{\otimes d}\) satisfies the property \(N_p\) for \(p \leq 3d -3\). Second we show that when \(\dim X = n \geq 3\) the bundle \(A^{\otimes d}\) satisfies the property \(N_p\) for \(p \leq d - n + 2\) and \(d \geq n - 1\). Toric varieties, Newton polyhedra, Okounkov bodies, Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) On higher syzygies of projective toric varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties Let \(X\) be a smooth algebraic surface with function field \(K\) and let \(\tau:V\to X\) be a standard \(\mathbb{P}^2\)-bundle over \(X\), i.e. \(\tau\) is a flat contraction morphism of an extremal ray of a smooth projective variety \(V\) with the generic fibre isomorphic to a \(K\)-form of \(\mathbb{P}^2\), i.e. \(V \times_X\overline K=\mathbb{P}^2\) for the algebraic closure \(\overline K\) of \(K\). In this paper, some birational maps from \(V\) to a standard \(\mathbb{P}^2\)-bundle \(W\) are represented by compositions of elementary birational morphisms, where \(W\) is a standard \(\mathbb{P}^2\)-bundle over the blow-up of \(X\) at a point of the non-smooth locus \(\Delta\) of \(\tau\). Let \(C\) be a smooth curve on \(X\) intersecting \(\Delta\) transversely at one point. A birational map from \(V\) to a standard \(\mathbb{P}^2\)-bundle over \(X\) which is isomorphic over \(X-C\), is decomposed into elementary birational morphisms. These are generalizations of the results about standard conic bundles by \textit{V. G. Sarkisov} [Math. USSR, Izv. 20, 355-390 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371-408 (1982; Zbl 0593.14034)]. function field; contraction; extremal ray; birational morphisms; standard conic bundles Birational automorphisms, Cremona group and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and birational maps, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Birational maps of standard projective plane bundles over algebraic surfaces

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties We give a corrected and strengthened statement and proof of the `\(p\)-adic analytic arc lemma' in a paper of the author [J. Lond. Math. Soc. (2) 73, No. 2, 367--379 (2006; Zbl 1147.11020)]. We show that the analytic arc is guaranteed to exist for \(p \geqslant 5\) and give a counterexample showing that this sometimes cannot be done when \(p = 2\). affine varieties; automorphisms; Skolem-Mahler-Lech theorem Bell, Corrigendum: ''A generalised Skolem-Mahler-Lech theorem for affine varieties'', J. London Math. Soc. 78 ((2)) pp 267-- (2008) Counting solutions of Diophantine equations, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), \(p\)-adic and power series fields Corrigendum: A generalised Skolem-Mahler-Lech theorem for affine varieties

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The authors formulate the effective field theory of a D-particle on orbifolds of \(T^4\) by a cyclic group as a gauge theory in a \(V\)-bundle over the dual orbifold. They argue that this theory admits Fayet-Iliopoulos terms analogous to those present in the case of non-compact orbifolds. In the \(n=2\) case, they present some evidence that turning on such terms resolves the orbifold singularities and may lead to a K3 surface realized as a blow up of the fixed points of the cyclic group action. effective field theory; D-particle; orbifolds; orbifold singularities; K3 surface; cyclic group action Greene, B. R.; Lazaroiu, C. I.; Yi, P.: D-particles on T4/zn orbifolds and their resolutions. Nucl. phys. B 539, 135-165 (1999) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, \(K3\) surfaces and Enriques surfaces, Relationships between surfaces, higher-dimensional varieties, and physics, Modifications; resolution of singularities (complex-analytic aspects) D-particles on \(T^4/\mathbb{Z}_n\) orbifolds and their resolutions

It is well known that singularities of a normal toric variety can be resolved by a regular decomposition of its fan. In this paper the authors consider an affine toric variety \(Z^\Gamma\) non necessarily normal embedded in an affine normal toric variety \(Z_{\rho}\). Then they give a method to resolve the singularities of \(Z_{\rho}\), this process will also resolve the singularities of \(Z^{ \Gamma }\). fan decomposition Pérez, PDG; Teissier, B, Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 379-382, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Embedded resolutions of non necessarily normal affine toric varieties The classical correspondence between Yang-Mills connections and stable bundles is generalized from orientable to non-orientable surfaces. The author considers non-orientable (real) surfaces having holomorphic or antiholomorphic transition functions. A related theory of \(d\)-holomorphic vector bundles is constructed for bundles over surfaces of this kind. The relevant concepts like Yang-Mills connection, Hodge star and others, make sense in this framework. Yang-Mills connections; stable bundles; non-orientable surfaces; \(d\)-holomorphic vector bundles Wang S., A Narasimhan-Seshadri-Donaldson correspondence over non-orientable surfaces, Forum Math., 1996, 8(4), 461--474 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Variational problems concerning extremal problems in several variables; Yang-Mills functionals, Finite transformation groups, Vector bundles on curves and their moduli A Narasimhan-Seshadri-Donaldson correspondence over non-orientable surfaces