text
stringlengths 571
40.6k
| label
int64 0
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 The authors present algorithms to compute the Cox rings of certain quotient singularities and of some closely related objects.
Precisely, consider a finite subgroup \(G \subset {\text{GL}}(n):={\text{GL}}(n,\mathbb C)\) not containing any pseudo-reflection and the quotient variety \(X_0={\mathbb C}^n/G\).
Firstly, they give an algorithm presenting the Cox ring \({\mathcal R}(X_0)\) as a quotient \({\mathbb C}[T_1, \ldots, T_s]/I_0\), where a set of generators of the ideal \(I_0\) is described. Now, \({{\mathcal R} (X_0})\) is a multi-graded ring. They also give a matrix that determines this multi-grading.
Next, they present an algorithm to construct a resolution \(X \to X_0\) of the singularities of \(X_0\) and to describe the multi-graded Cox ring \({\mathcal R} (X)\). More precisely, the idea is to construct an embedding of \(X_0\) into a toric variety \(Z_0\) and a modification \(Z \to Z_0\) which is a resolution of the singularities of \(Z_0\). Letting \(X\) denote the strict transform of \(X_0\) to \(Z\), they give a description of \({\mathcal R} (X)\) as a quotient of a polynomial ring by an ideal whose generators are specified, they also obtain a matrix defining the multi-grading. This variety \(X\) is a ``candidate'' to be a resolution of the singularities of \(X\). Indeed, if \(X\) is smooth, then it is a resolution of \(X_0\). The variety \(X\) might be singular, but they give an algorithm to decide whether \(X\) is smooth or not.
In the construction, among other methods they use techniques from toric and tropical geometries.
Although the algorithms are rather complicated, by using the Computer Algebra System \textbf{Singular} the authors are able to implement it in a number of concrete examples. For instance, they do it for subgroups \(G \subset {\text{GL}}(3)\) of order \(\leq 12\), and they study some interesting properties of the resulting resolutions (e.g., to decide when they are crepant).
In the final section they implement the algorithm in two examples involving \( {\text{GL}}(4)\), although it seems that in this case (\(n=4\)) a more general implementation requires too many computations for the presently available machines. Cox ring; resolution of singularities; quotient singularity; algorithm; toric variety; tropical variety Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of higher-dimensional varieties, Geometric invariant theory, Toric varieties, Newton polyhedra, Okounkov bodies, Special varieties Computing resolutions of quotient singularities | 0 |
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 article is to describe a representation of the \(A\)-resultant as a single determinant, and to express this resultant as a polynomial in \(3\times 3\) determinants. This representation is a generalization of the classical Dixon resultant for three bivariate polynomials of bidegree \((m,n)\).
Using \(A\)-resultants we can find implicit equations for Hirzebruch surfaces \(F_n, n\in \mathbb{N}\), cyclides, Warren hexagons and toric pentagons. The algorithm has some advantages. It is fast, simple and for the mentioned surfaces can be easily computed using for example the MAPLE package. \(A\)-resultant; Dixon resultant; Hirzebruch surfaces; Warren hexagons; toric pentagons; MAPLE Zub\doteq, S.: The n-sided toric patches and A-resultants. Computer aided geometric design 17, 695-714 (2000) Computational aspects of algebraic surfaces, Computer science aspects of computer-aided design, Computer-aided design (modeling of curves and surfaces), Low codimension problems in algebraic geometry The \(n\)-sided toric patches and \(\mathcal A\)-resultants | 0 |
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 a complex projective variety of positive dimension \(n\) with isolated singularities, \(\pi: X \rightarrow Y\) a resolution of singularities, and \(G = \pi^{-1}\text{Sing}(Y)\) the exceptional locus. It is well-known that in this case the natural morphism \(\gamma: H^{k-1}(G)\rightarrow H^k(Y,Y\setminus \text{Sing}(Y))\) vanishes for \(k>n\). However, the same is also true when either \(\dim G <n/2\), or when \(Y\) is normal variety, or when \(\pi\) is the blowing-up of \(Y\) along the singular locus with smooth and connected fibres, or when \(\pi\) admits a natural Gysin morphism, etc. The authors explain that the vanishing condition is, in fact, equivalent to the Decomposition Theorem valid in the bounded derived category of sheaves of \(\mathbb Q\)-vector spaces on \(Y\) in the sense of [\textit{G. Williamson}, in: Séminaire Bourbaki. Volume 2015/2016. Exposés 1104--1119. Avec table par noms d'auteurs de 1948/49 à 2015/16. Paris: Société Mathématique de France (SMF). 335--367, Exp. No. 1115 (2017; Zbl 1373.14010)]. As a result, they obtain a short and simple proof of this theorem in all cases when \(\gamma\) vanishes, discuss some useful relations with topological bivariant theory, and so on. projective variety; isolated singularities; resolution of singularities; derived category; intersection cohomology; decomposition theorem; bivariant theory; Gysin morphism; cohomology manifold Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Sheaves in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds On the topology of a resolution of isolated singularities | 0 |
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 say that an exact equivalence between the derived categories of two algebraic varieties is tilting type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories Deformation of tilting-type derived equivalences for crepant resolutions | 0 |
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 apply the results of \textit{Z. Wang} [The twisted Mellin transform, electronic only, available at \url{http://arxiv.org/abs/0706.2642v2}] on the twisted Mellin transform to problems in toric geometry. In particular, we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, \(z^{k}, k \in \mathbb{Z}^{d}\), in Bargmann space and prove an ``upstairs'' version of the spectral density theorem of \textit{D. Burns, V. Guillemin} and \textit{A. Uribe} [The spectral density functions of a toric variety, electronic only, available at \url{http://arxiv.org/abs/0706.3039v1}]. We also obtain for the \(z^{k}\)'\(s\), ``upstairs'' versions of the results of \textit{B. Shiffman, T. Tate} and \textit{S. Zelditch} [Ann. Inst. Fourier 54, No.~5, 1497--1546 (2004; Zbl 1081.35063)] on distribution laws for eigenstates on toric varieties. Guillemin, V.; Wang, Z.: The Mellin transform and spectral properties of toric varieties, Transform. groups 13, No. 3-4, 575-584 (2008) Toric varieties, Newton polyhedra, Okounkov bodies, Global differential geometry of Hermitian and Kählerian manifolds The Mellin transform and spectral properties of toric varieties | 0 |
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 earlier work with \textit{V. Uma} [Osaka J. Math. 44, No.~1, 71--89 (2007; Zbl 1125.55003)], the author gave a description of the \(K\)-ring of a quasi-toric manifold in terms of generators and relations. The purpose of this paper is to give a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope (Theorem 5.3).
Using a method presented in the above paper, \textit{V. Uma} has previously shown in [Contemporary Mathematics 460, 385--389 (2008; Zbl 1151.55005)] that the same result as that obtained here holds under the additional assumption that these torus manifolds satisfy a shellability condition. The author presents here an approach to avoid this shellability condition.
The paper consists of two parts, the first of which gives a description of the \(K\)-ring of a non-singular complete toric variety in terms of generators and relations (Theorem 2.2). In the second part, it is shown that one can obtain the same result for a torus manifold by carrying out exactly all the steps of the proof corresponding to the above case. Then one uses results of \textit{M. Masuda} and \textit{T. Panov} [Osaka J. Math. 43, No.~3, 711--746 (2006; Zbl 1111.57019)]. But in fact, the second part subsumes the first, since the class of torus manifolds includes complete nonsingular toric varieties as well as quasi-toric manifolds.
The result obtained can be summarized as follows. Let \(X\) be a smooth compact oriented connected manifold with an effective smooth action of a torus \(T\) such that \(X^T\neq \phi\). From a local standardness condition given one finds that there are only finitely many codimension 2 submanifolds \(V_1, \dots, V_d\), each of which is pointwise fixed by a circle subgroup of \(T\). These circle subgroups detemine \(v_1, \dots, v_d \in \text{Hom}({\mathbb{S}}^1, T)=H_2(BT)\). Let \(Q_i\) denote the image of \(V_i\) under the quotient map from \(X\) to \(Q=X/T\). (This orbit space becomes a homology polytope with \(d\) facets \(Q_1, \dots, Q_d\).) Then it follows that \(K(X)\) is isomorphic to the ring \(\mathbb{Z}[y_1, \dots, y_d]/{\mathcal I}\), where \({\mathcal I}\) is the ideal generated by the elements \(y_{j_1}, \dots, y_{j_k}\), whenever \(\bigcap_{i=1}^kQ_j=\phi\) in \(Q\), and
\[
\prod_{j, \langle u, v_j \rangle >0}(1-y_j)^{\langle u, v_j \rangle} -\prod_{j, \langle u, v_j \rangle <0}(1-y_j)^{-\langle u, v_j \rangle}
\]
for \(u \in H^2(BT)\). \(K\)-theory; smooth complete toric varieties; torus manifolds Topological \(K\)-theory, Toric varieties, Newton polyhedra, Okounkov bodies \(K\)-rings of smooth complete toric varieties and related spaces | 0 |
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 notion of a \(k\)-configuration was introduced for points in \(\mathbb{P}^2\) by \textit{L. G. Roberts} and \textit{M. Roitman} [J. Pure Appl. Algebra 56, 85-104 (1989; Zbl 0673.13015)] and generalized to points in \(\mathbb{P}^3\) by \textit{T. Harima} [J. Pure Appl. Algebra 103, 313-324 (1995; Zbl 0847.13003)]. In the former case the Hilbert function and minimal free resolution were both known, while in the latter case the Hilbert function was known. The present paper computes the minimal free resolution of the homogeneous ideal of a \(k\)-configuration in \(\mathbb{P}^3\). The authors give a set of parameters for \(k\)-configurations and show that the Betti numbers of a particular \(k\)-configuration depend only on its parameters. Furthermore, they show that for any given set of parameters there exists a set of points \(X\) which is a \(k\)-configuration, and a linear form \(L\in S= k[X_0,\dots, X_3]\) which is not a zero-divisor on \(S/I_X\), such that if \(J= (I_X,L)/ (L)\) in \(R= S/(L)\) then \(J\) is a lex-segment ideal. It then follows from work of \textit{A. M. Bigatti} [Commun. Algebra 21, 2317-2334 (1993; Zbl 0817.13007)] and \textit{H. A. Hulett} [ibid., 2335-2350 (1993; Zbl 0817.13006)] that all \(k\)-configurations in \(\mathbb{P}^3\) in fact have extremal Betti numbers. This work has been generalized to any projective space \(\mathbb{P}^n\) in a joint work of the current authors and Harima [\textit{A. V. Geramita, T. Harima} and \textit{Y. S. Shin}, Queen's Pap. Pure Appl. Math. 114, 97-140, Exposé II D (1998; Zbl 0943.13012)]. Hilbert function; minimal free resolution of the homogeneous ideal of a \(k\)-configuration A. V. Geramita and Y. S. Shin, \(k\)-configurations in \(\mathbb{P}^3\) all have extremal resolutions, J. Algebra 213 (1999), 351--368. Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) \(k\)-configurations in \(\mathbb{P}^3\) all have extremal resolutions | 0 |
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 Hoşten investigates a question of Batyrev about the maximal number of primitive collections in a complete regular fan with given codimension \(n-d\). A primitive collection is a minimal subset of the set \(A\) of one-dimensional generators that does not form a cone of the fan.
The idea is to combine the following ingredients:
(i) A fan may be considered as a certain triangulation of \(A\), and \textit{L. J. Billera, I. M. Gelfand} and \textit{B. Sturmfels} [J. Comb. Theory, Ser. B 57, No. 2, 258-268 (1993; Zbl 0727.05018)] have shown that each coherent triangulation of \(A\) corresponds to a chamber in the equally named complex \(\Gamma(B)\) built from the Gale transform \(B\) of \(A\).
(ii) Adjacent chambers correspond to triangulations differing by a bistellar flip.
(iii) There is an injection from the set of circuits of a fan (i.e.\ data that give rise to a bistellar flip) into the set of primitive collections. In particular, Hoşten obtains the number of facets of a chamber in \(\Gamma(B)\) as a lower bound of the number of primitive collections in the corresponding triangulation of \(A\).
He uses this result to present a special fan (arising from the complete bipartite graph \(K_{2k-1,2k+1}\)) in dimension \(d=4k(k-1)\) with \(n=(2k-1)(2k+1)\) one-dimensional generators and \(4^k\) primitive collections. In particular, this establishes an example of a fan with codimension \(n-d\) with more than \(2^{(n-d)/2}\) primitive collections.
Finally, Hoşten remarks that the result may be reformulated in terms of degenerations to monomial ideals. His fan gives rise to a toric ideal such that the reduced Gröbner basis with respect to the degree lexicographic term order has at least \(2^{(n-d)/2}\) elements. toric varieties; complete regular fan; bistellar flip; number of facets of a chamber; degenerations of monomial ideals; toric ideal; Gröbner basis Toric varieties, Newton polyhedra, Okounkov bodies, Gale and other diagrams, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) On the complexity of smooth projective toric varieties | 0 |
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 a variant of Shokurov's criterion of toric surface, we give a criterion of two new classes of normal projective surfaces, called pseudotoric surfaces of defect one and half-toric surfaces. A typical example of pseudo-toric surface of defect one is a projective toric surface blown up at a non-singular point of the boundary divisor. A half-toric surface is the quotient of a projective toric surface by an almost free involution preserving the boundary divisor. The structure of pseudo-toric surface of defect one and that of half-toric surface are also studied in detail. normal surface; rational surface; toric variety Rational and ruled surfaces, Toric varieties, Newton polyhedra, Okounkov bodies, Families, moduli, classification: algebraic theory A variant of Shokurov's criterion of toric surface | 0 |
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 construct a differential graded Lie algebra \(\mathrm{Der}_k(R,R)\) controlling infinitesimal flat deformations of a separated \(k\)-scheme \(X\) where \(k\) is a field of characteristic \(0\). This is an application of the homotopy-theoretic deformation theory they develop in [\textit{M. Manetti}: ``Formal deformation theory in left-proper model categories'', Preprint, \url{arXiv:1802.06707}] and translates Palamodov's resolvent of a complex space into the algebraic category. Bases of deformations are non-positive local Artin dg algebras \(A \in \mathbf{DGArt}^{\leq 0}_k\), a derived analogue of classical local Artin algebras which are recovered as dg algebras concentrated in degree \(0\). In case \(X = \mathrm{Spec}\ S\) is affine (treated in the above mentioned work), cofibrant deformations \(A \to S_A\) of \(k \to S\) up to weak equivalence in the model category \(\mathbf{CDGA}^{\leq 0}_k\) form a (derived) deformation functor
\[
\mathrm{Def}_S: \mathbf{DGArt}^{\leq 0}_k \to \mathbf{Set}
\]
which generalizes flat deformations of \(S\). Taking a cofibrant resolution \(R \to S\), the deformation functor \(\mathrm{Def}_S\) is controlled by the dgla \(\mathrm{Der_k(R,R)}\) of derivations of \(R\). The authors call \(R \to S\) a (local) \textit{Tate-Quillen resolution} as it is the dg analogue of Quillen's simplicial resolution of rings and a generalization of the Koszul-Tate resolution. The dgla \(\mathrm{Der}_k(R,R)\) corresponds to the tangent complex in complex geometry.
The paper under review treats the general, non-affine case. Here the model category \(\mathbf{CDGA}^{\leq 0}_k\) is replaced by the (Reedy) model category
\[
\mathbf{M} := \mathrm{Fun}(\mathcal{N}, \mathbf{CDGA}^{\leq 0}_k)
\]
of diagrams of shape \(\mathcal{N}\) where \(\mathcal{N}\) is the nerve of an affine covering \(\{\mathcal{U}_\alpha\}\) of \(X\). The algebra \(S\) is replaced by the diagram \(S_\bullet \in \mathbf{M}\) of sections \(S_\alpha = \Gamma(U_\alpha,\mathcal{O}_X)\) on the opens from which \(X\) can be recovered by gluing. For a dg Artin algebra \(A \in \mathbf{DGArt}^{\leq 0}_k\), cofibrant deformations \(\Delta A \to S_{A\bullet}\) over the constant diagram \(\Delta A\) up to weak equivalence yield a deformation functor
\[
\mathrm{Def}_{S_\bullet}: \mathbf{DGArt}^{\leq 0}_k \to \mathbf{Set}
\]
which generalizes flat deformations of \(X\). Again a cofibrant resolution \(R_\bullet \to S_\bullet\) in \(\mathbf{M}\) provides a dgla controlling the deformation functor. The authors call such a cofibrant resolution a \textit{Reedy-Palamodov resolvent}. Whereas the first name refers to the use of the direct Reedy category \(\mathcal{N}\) and the Reedy model structure on \(\mathbf{M}\), the second honours Palamodov's globalization of Tyurina's semifree resolution of an analytic algebra by means of polyhedral coverings which yields the tangent complex and the tangent cohomology of a complex space. model categories; deformation theory; differential graded algebras; algebraic schemes; cotangent complex Formal methods and deformations in algebraic geometry, Graded rings and modules (associative rings and algebras), Homotopical algebra, Quillen model categories, derivators, Deformations of complex structures Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions | 0 |
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 To every toric ideal one can associate an oriented matroid structure, consisting of a graph and another toric ideal, called bouquet ideal. The connected components of this graph are called bouquets. Bouquets are of three types; free, mixed and non-mixed. We prove that the cardinality of the following sets - the set of indispensable elements, minimal Markov bases, the Universal Markov basis and the Universal Gröbner basis of a toric ideal - depends only on the type of the bouquets and the bouquet ideal. These results enable us to introduce the strongly robust simplicial complex and show that it determines the strongly robust property. For codimension 2 toric ideals, we study the strongly robust simplicial complex and prove that robust implies strongly robust. toric ideals; Graver basis; universal Gröbner basis; Markov bases; robust ideals Commutative rings defined by binomial ideals, toric rings, etc., Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Applications of graph theory, Algebraic statistics On the strongly robust property of toric ideals | 0 |
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 0632.00003.]
A torus embedding is a normal algebraic variety which possesses an effective action of a split algebraic torus with an open orbit. The author has previously studied reduced subschemes, called toric polyhedra, of a torus embedding which are partial unions of the orbits of the torus action. He constructed dualizing complexes of affine toric polyhedra consisting of coherent sheaves and gave criteria for the schemes to be Gorenstein or Cohen-Macaulay [see \textit{M.-N. Ishida}, Tôhoku Math. J., II. Ser. 32, 111-146 (1980; Zbl 0454.14021) and \textit{M.-N. Ishida} and \textit{T. Oda}, ibid. 33, 337-381 (1981; Zbl 0456.14005)].
In the present article, the author generalizes this complex for semi- toroidal varieties, that is, varieties locally isomorphic to toric polyhedra in the étale topology. Using this dualizing complex the author defines the de Rham complex, \({\tilde \Omega}_ X^{\bullet}\), of a semi-toroidal variety X with filtration. This de Rham complex is a generalization of that of \textit{V. Danilov} [Russ. Math. Surv. 33, 97-154 (1978); translation from Usp. Mat. Nauk 33, No.2(200), 85-134 (1978; Zbl 0425.14013)] and is proved to be equal to the de Rham complex of du Bois for these varieties [\textit{P. du Bois}, Bull. Soc. Math. Fr. 109, 41-81 (1981; Zbl 0465.14009)]. In particular, if X is complete, the natural spectral sequence, \(E_ 1^{p,q}=H\) q(X,\({\tilde \Omega}\) \(p_ X)\Rightarrow H^{p+q}(X,{\mathbb{C}})\), degenerates at the \(E_ 1\)-terms and converges to the Hodge filtration. torus embedding; toric polyhedra; semi-toroidal varieties; dualizing complex; de Rham complex Masa-Nori Ishida, Torus embeddings and de Rham complexes, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 111 -- 145. Group actions on varieties or schemes (quotients), de Rham cohomology and algebraic geometry Torus embeddings and de Rham complexes | 0 |
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 survey the problems of resolution of singularities in positive characteristic and of local and global monomialization of algebraic mappings. We discuss the differences in resolution of singularities from characteristic zero and some of the difficulties. We outline Hironaka's proof of resolution for positive characteristic surfaces, and mention some recent results and open problems.
Monomialization is the process of transforming an algebraic mapping into a mapping that is essentially given by a monomial mapping by performing sequences of blow ups of nonsingular subvarieties above the target and domain. We discuss what is known about this problem and give some open problems. Cutkosky, Steven Dale, Ramification of valuations and counterexamples to local monomialization in positive characteristic, (2014), preprint Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Resolution of singularities in characteristic \(p\) and monomialization | 0 |
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 present paper is a tutorial on toric varieties. It explains the general theory, in particular the real one, by presenting a lot of examples and pictures.
In particular, the author shows how toric varieties are naturally involved in the subject of geometric modeling: Via linear projections, toric varieties provide easy parametrizations of rational varieties -- even if their implicit equations are rather complicated. Moreover, the moment map appears in this context as an extremal projection of the positive real part of a toric variety. toric varieties; geometric modeling; real toric structures F. Sottile, \textit{Toric ideals, real toric varieties, and the algebraic moment map}, \textit{Contemp. Math.}\textbf{334} (2003) 225 [math.AG/0212044]. Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computer-aided design (modeling of curves and surfaces), Computer graphics; computational geometry (digital and algorithmic aspects), Computer science aspects of computer-aided design Toric ideals, real toric varieties, and the moment map | 0 |
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 of characteristic zero and let \(W\) be a vector space over \(K\) of dimension \(N+1\). Let \(E=\bigwedge(W^*)\) be the exterior algebra on the dual of \(W\) and let \(S\) be the symmetric algebra on \(W\) with the usual grading. The Bernstein-Gel'fand-Gel'fand correspondence asserts that there is an equivalence of categories, the derived category of bounded complexes of finitely generated \(S\)-modules and the category of graded free \(E\)-modules [\textit{I. N. Bernshtejn, I. M. Gel'fand} and \textit{S. I. Gel'fand}, Funkts. Anal. Prilozh. 12, No. 3, 66--67 (1978; Zbl 0402.14005)]. \textit{D. Eisenbud, G. Fløystad} and \textit{F.-O. Schreyer} showed that the essential part of the BGG correspondence is given via the Tate resolution [Trans. Am. Math. Soc. 355, No. 11, 4397--4426 (2003; Zbl 1063.14021)]. These are related to the Weyman complexes [\textit{J. M. Weyman}, Cohomology of vector bundles and syzygies. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)].
In the paper under review, the authors describe a generalization of Weyman complexes and describe an explicit formula for the differentials in this complex in terms of differentials in the corresponding Tate resolution. They also show that these complexes are in fact certain Fourier-Mukai transforms. Tate resolutions; Weyman complexes; Fourier-Mukai transforms Cox, D.; Materov, E.: Tate resolutions and Weyman complexes, Pacific J. Math. 252, No. 1, 51-68 (2011) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Syzygies, resolutions, complexes and commutative rings Tate resolutions and Weyman complexes | 0 |
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 necessary and sufficient condition for the isomorphic projection of a \(k\)-normal variety to remain \(k\)-normal, \(k=2\); the condition is based on a scheme \(Z_k\) naturally associated to degree \(k\) forms vanishing on the variety. We furnish many applications and examples especially in the case of varieties defined by quadratic equations. A non-vanishing theorem for the Koszul cohomology of projected varieties allows us to construct interesting examples in the last sections. All the results are effective and also interesting from the computational point of view. Alzati A. and Russo F. (2002). On the k-normality of projected algebraic varieties. Bull. Braz. Math. Soc. (N.S.) 33(1): 27--48 Projective techniques in algebraic geometry, \(n\)-folds (\(n>4\)) On the \(k\)-normality of projected algebraic varieties. | 0 |
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 Igusa's strong monodromy conjecture states that the real part of poles of the \(p\)-adic zeta function associated to a hypersurface defined by a polynomial \(f\) with integer coefficients determine roots of the Bernstein-Sato polynomial of \(f\). A consequence of this conjecture is a link between poles of Igusa's zeta function and monodromy eigenvalues of the Milnor fiber of \(f\).
Some generalizations of Igusa's strong monodromy conjecture were proposed thank to the work of Denef and Loeser (who introduced their motivic zeta function using motivic integration) and \textit{N. Budur} et al. [Compos. Math. 142, No. 3, 779--797 (2006; Zbl 1112.32014)] (who defined a generalization of Bernstein-Sato polynomial \(b_I^X\) for an ideal \(I\) on a smooth affine variety \(X\)).
In this paper, the conjecture is tested for a monomial ideal \(I\) on an affine toric variety \(U_\sigma\). One of the key ingredients for this is an explicit formula for the motivic zeta function. This description is obtained understanding the geometry of the contact locus of a monomial ideal on a toric variety and using a change of variables formula provided in [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004)] applied to a toric log-resolution.
The motivic zeta function the authors construct has some similarities to the case studied in [\textit{J. Howald} et al., Proc. Am. Math. Soc. 135, No. 11, 3425--3433 (2007; Zbl 1140.14003)], except that it may have poles coming from rays of the fan \(\Delta\) that are not rays of the normal fan \(\Delta_I\). Some examples are provided where such rays produce poles of the motivic zeta function that are not roots of the Bernstein-Sato polynomial. However, it is also proven that, under some hypothesis, the poles introduced by rays of the normal fans determine roots of the Bernstein-Sato polynomial. Bernstein-Sato polynomials; motivic zeta functions; strong monodromy conjecture; toric varieties Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry, Arcs and motivic integration, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials The strong monodromy conjecture for monomial ideals on toric varieties | 0 |
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 problem of resolution of singularities -- solved by Hironaka about half a century ago over a field of characteristic 0 -- is still open in full generality. For arithmetical schemes \(\mathcal X\), the general case of \(\dim \mathcal{X} >2 \) remained unsolved until now. The next step is done here by the authors with their
Theorem 1.1. Let \(\mathcal X\) be a reduced and separated Noetherian scheme which is quasi-excellent and of dimension at most 3. There exists a proper birational morphism \(\pi :\mathcal{X}'\to\mathcal{X}\) with the following properties:
\begin{enumerate}
\item[(i)] \(\mathcal{X}'\) is everywhere regular;
\item[(ii)] \(\pi\) induces an isomorphism \(\pi^{-1} (\mathrm{Reg} \ \mathcal{X}) \to\mathrm{Reg}\mathcal{X} \);
\item[(iii)] \(\pi^{-1} (\mathrm{Sing } \ \mathcal{X})\) is a strict normal crossings divisor on \(\mathcal{X}'\).
\end{enumerate}
If furthermore a finite affine covering \(\mathcal{X} = \mathcal{U}_1 \cup \mathcal{U}_2 \cup \dots \cup \mathcal{U}_n \) is specified, one may take \(\pi^{-1}(\mathcal{U}_i) \to \mathcal{U}_i\) projective, \(1\leq i \leq n\).
A proper birational morphism \(\pi\) as above is said to be a resolution if it satisfies properties (i) and (ii) above. If additionally (iii) is satisfied, \(\pi\) is said to be a good resolution.
It is pointed out that the construction of \(\pi\) is not given as a sequence of Hironaka-permissible blowing ups. The authors give evidence for situations, when this can be achieved proving a local version of the theorem which uses only Hironaka-permissible blowing ups. An answer to the question whether this could hold in general is referred to as \textit{widely open}.
The following can be deduced from the theorem.
Corollary 1.2. Let \(A\) be a reduced complete Noetherian local ring of dimension three. Then \(\mathcal{X} := \mathrm{Spec} \ A\) has a good resolution of singularities which is projective.
Corollary 1.3. Let \(\mathcal O\) be an excellent Dedekind domain with quotient field \(F\) and \(\Sigma / F\) be a regular projective surface. There exists a proper and flat \(\mathcal O\)-scheme \(\mathcal X\) with generic fiber \(\mathcal{X}_F = \Sigma\) which is everywhere regular.
In this paper the authors continue their earlier work on resolution of singularities of threefolds. In the first sections they start developping a more general approach to the problem for hypersurface singularities defined by a reduced polynomial \(h=X^p+f_1X^{p-1}+ \dots + f_p \in S[X], f_i\in S\) over an excellent regular local ring \(S\) of any dimension \(\geq 1\), where \(p:= \mathrm{char} (S/\mathbf{m}_S) >0\). It is supposed -- using the notations \(K:= Q(S)\), \(\mathcal{X}:= \mathrm{Spec} (S[X]/(h))\) and \(L\) for the total quotient ring of \(S[X]/(h)\) -- that
\begin{enumerate}
\item[(i)] \(\mathrm{char} (K)=p\) and \(f_i =0\) for \(i<p\), or
\item[(ii)] \(\mathcal{X}\) is \(G\)-invariant, where \(G:= \mathrm{Aut}_K(L) = \mathbb{Z}/(p)\).
\end{enumerate}
Under these assumptions and for \(\mathrm{dim} (S)=3\), the main part of the article gives the following version of a resolution as stated in
Theorem 1.5. Let \(\mu\) be a valuation of \(L\) which is centered in \(\mathbf{m}_S\). There exists a composition of local Hironaka-permissible blowing ups \((\mathcal{X}=: \mathcal{X}_0,x_0 ) \leftarrow (\mathcal{X}_1,x_1 ) \leftarrow \dots \leftarrow (\mathcal{X}_r,x_r ) \), where \(x_i\in \mathcal{X}_i\) is the center of \(\mu\), such that \((\mathcal{X}_r,x_r) \) is regular.
Main combinatorial tool is a variant of Hironaka's characteristic polyhedron attached with the singularity. The inductive procedure is controlled by a numerical function which is different from the ``classical'' pair of multiplicity and slope function used for hypersurface singularities in residue characteristic 0.
Applying an idea which can be traced back to Zariski, Theorem 1.5 implies the above Theorem 1.1. resolution of singularities; arithmetical varieties; Zariski; blowing up; valuations Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Varieties over finite and local fields, Varieties over global fields Resolution of singularities of arithmetical threefolds | 0 |
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 Makar-Limanov invariant was introduced just over 10 years ago, and has emerged as one of the more important new tools in the study of affine varieties. The paper under review studies a class of surfaces \(V\) with trivial Makar-Limanov invariant, meaning that \(V\) admits two independent \(\mathbb{C}^+\)-actions. In particular, this condition implies that \(V\) has a relatively large group of automorphisms. The paper's main result is the following
Theorem. A normal affine surface \(V\) that is nonisomorphic to \(\mathbb{C}^*\times \mathbb{A}^1\) has a trivial Makar-Limanov invariant if and only if \(V\) is completable by a zigzag.
This generalizes earlier work of \textit{T. Bandman} and \textit{L. Makar-Limanov} [Mich. Math. J. 49, No. 3, 567--582 (2001; Zbl 1079.14539)], \textit{J. Bertin} [J. Reine Angew. Math. 341, 32--53 (1983; Zbl 0501.14028)], and \textit{M. H. Gizatullin} [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1047--1071 (1971; Zbl 0221.14023)], and is closely related to recent results of \textit{D. Daigle} and \textit{P. Russell} [Can. J. Math. 56, No. 6, 1145--1189 (2004; Zbl 1073.14075)], and \textit{M. Miyanishi} and \textit{R. V. Gurjar} [On the Makar-Limanov invariant and fundamental group at infinity, preprint (2002)]. Overall, this is a very well-written paper, drawing on standard tools from the theory of algebraic surfaces to give a clear and largely self-contained exposition of its results. A. Dubouloz, Generalized Danielewski surfaces, preprint, prépublication de l'Institut Fourier n.612, 2003. Classification of affine varieties, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Affine fibrations Completions of normal affine surfaces with a trivial Makar-Limanov invariant | 0 |
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 main tool for studying the inflections (or Weierstrass points) of a mapping of a smooth projective variety into projective space are the principal parts of line bundles. In recent work by \textit{D. A. Cox} [J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)], homogeneous coordinates on a toric variety have been introduced, and in subsequent work [\textit{V. V. Batyrev} and \textit{D. A. Cox}, Duke Math. J. 75, No. 2, 293-338 (1994; Zbl 0851.14021)] an Euler sequence is defined. The homogeneous coordinates and the Euler sequence are direct generalizations of the usual notions in the case of projective space. The purpose of this note is to use the Euler sequence to describe the principal parts of line bundles on a toric variety (theorem 1.2). The essential idea is to compare derivatives with respect to local and global coordinates. inflections; Weierstrass points; homogeneous coordinates; Euler sequence; toric variety Perkinson, D.: Principal parts of line bundles on toric varieties. Compos. Math. 104(1), 27--39 (1996) Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Principal parts of line bundles on toric varieties | 0 |
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 article under review completes a project envisioned by a group including the authors regarding projective modules over real affine varieties. One of the earliest papers in this direction is by \textit{S. M. Bhatwadekar} and \textit{R. Sridharan} [Invent. Math. 136, No. 2, 287--322 (1999; Zbl 0949.14005)]. The question being dealt with is the following:
If \(A\) is a smooth affine algebra over the reals of dimension \(n\geq 2\) and \(P\) is a projective module over \(A\) of rank \(n\) with its \(n\)-th Chern class \(c_n(P)=0\) what further restrictions are necessary for \(P\) to split off a free direct summand?
The question was answered by \textit{M.~P.~Murthy} [Ann. Math. (2) 140, No. 2, 405--434 (1994; Zbl 0839.13007)] when \(A\) is an affine algebra over an algebraically closed field, deciding that no further restrictions were necessary. Of course over the reals, one does need further restrictions. In the above mentioned paper of Bhatwadekar and Sridharan, they proved that the vanishing of the \(n\)-th Chern class is sufficient even over the reals if \(n\) is odd and the canonical bundle of \(A\) as well as the determinant of \(P\) are trivial. In this paper, the authors give a complete and satisfactory answer to the above question. affine algebras; projective modules; Euler class S. M. Bhatwadekar, M. K. Das and Satya Mandal, Projective modules over smooth real affine varieties, Invent. Math., 136 (2006), 151--184. Projective and free modules and ideals in commutative rings, Real algebraic and real-analytic geometry Projective modules over smooth real affine varieties | 0 |
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 \(S\) be a normal projective algebraic surface over the complex numbers with only quotient singularities. It is known that the fundamental group of the smooth part \(S_0\) of \(S\) is finite if the anti-canonical divisor \(- K_S\) of \(S\) is big and nef. But if \(- K_S\) is only big, \(\pi_1 (S_0)\) may be infinite as shown by examples in this paper. Furthermore, there is a kind of classification of surfaces \(S\) with \(- K_S\) big or with anti-Kodaira dimension \(\kappa (- K_S) = 1\) and \(\pi_1 (S_0)\) infinite. non-finiteness of fundamental group; anti-canonical divisor; classification of surfaces D.-Q. Zhang, Normal algebraic surfaces of anti-Kodaira dimension one or two, Intern. J. Math. 6 (1995), 329--336. Homotopy theory and fundamental groups in algebraic geometry, Divisors, linear systems, invertible sheaves Normal algebraic surfaces of anti-Kodaira dimension one or two | 0 |
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 nicely written paper the authors study the standard generalized Gorenstein algebras of homological dimension three. Let \(R = \Bbbk[x_{1},\dots,x_{r}]\) with \(r\geq 3\) and \(\Bbbk\) is any field. A graded standard \(R\)-algebra \(R/I\) of homological dimension \(3\) is called generalized Gorensten algebra if the rank of the last syzygy module is \(1\) in its minimal graded free resolution. In particular, the authors investigate graded minimal resolutions of the type \[0 \rightarrow R(-s) \rightarrow \bigoplus_{j=1}^{n}R(-b_{j}) \rightarrow \sum_{i=1}^{n}R(-a_{i}) \rightarrow R.\] Now we say that a matrix \(M \in R^{n,m}\) is a presentation matrix if it is associated to a map \(\phi\) in a presentation of the form \[R^{m} \xrightarrow{\phi} R^{n} \rightarrow R.\] Now if \(M \in R^{n,m}\), \(n\leq m\), with rank \(\mathrm{rk }\, M = n-1\), and \(N\) is a submatrix of \(M\) of size \(n\times n-1\) of rank \(n-1\), then we define \(g_{i}(M) =g_{i}(N)/d_{N}\), where \(g_{i}(N)\) is the minor of \(N\) obtained by deleting the \(i\)-th row and \(d_{N} = \mathrm{GCD}(g_{1}(N),\dots, g_{n}(N))\). We set additionally \(\gamma(M) = (g_{1}(M), \dots, g_{n}(M))\). Observe that \(\gamma(M)\) generates an ideal \(I_{M}\) with \(\mathrm{depth} \, I_{M} \geq 2\). Moreover, \(\gamma(M)\) defines a map \(\gamma(M) : R^{n} \rightarrow R\). One can show that if \(M \in R^{n,n}\) is a square presentation matrix, that \(R/I_{M}\) is a generalized Gorenstein algebra whose free resolution is of the form \[0 \rightarrow R^{*} \xrightarrow{\gamma(\phi)^{*}} R^{n} \xrightarrow {\phi} R^{n} \xrightarrow{\gamma(\phi)} R \rightarrow R/I_{M} \rightarrow 0,\] where \(\phi\) is the map associated with the matrix \(M\). If the above resolution is minimal, than we say that \(M\) is a minimal presentation matrix.
The main structural result of the paper can be formulated as follows.
Theorem A. Let \(M \in R^{n,n}\) be a matrix of rank \(n-1\), \(\gamma(M) = (g_{1}, \dots, g_{n})\), \(\gamma(M^{T}) = (h_{1}, \dots, h_{n})\), and let \(J\) be the ideal generated by \(\gamma(M^{T})\). Denote by \(M^{C}\) the cofactor of matrix \(M\). The matrix \(M\) is a presentation matrix if and only if \(\mathrm{depth} \, J \geq 3\) and \(M^{C} =[u(g_{i}h_{j})]\), where \(u\) is a unit.
Theorem B. Let \(X \subset \mathbb{P}^{r}\) with \(r\geq 3\) be a closed projective scheme whose defining ideal \(I_{X}\) has a graded minimal free resolution of the form \[0 \rightarrow R \xrightarrow{\rho = (\tau,0)} R^{3} \oplus R^{n-3} \xrightarrow{(\alpha \kappa) \oplus \delta} R^{n} \rightarrow R,\] where \(\tau(1) = (h_{1},h_{2},h_{3})\) and \((h_{1},h_{2},h_{3})\) is a regular sequence, \(\kappa: R^{3} \rightarrow R^{3}\) is the Koszul map on \(h_{1},h_{2},h_{3}\), \(\alpha : R^{3} \rightarrow R^{n}\), \(\delta:R^{n-3} \rightarrow R^{n}\) are suitable maps. Let \(Z\) be the complete intersection defined by \(I(\rho) = I(\tau)\), and let \(S = V(\mathrm{det}(\alpha \oplus \delta))\). Denote by \(Y\) the scheme defined by \(I(\alpha \oplus \delta, \rho^{*})\). If \(\mathrm{codim} \, (S\cap Z) =4\), then \(X = Y \cup Z\).
Next, the authors focus on the case \(n=3\) providing an explicit characterization of the graded Betti numbers for generalized Gorenstein ideals having a graded minimal free resolution of the type \[0 \rightarrow F_{3} \xrightarrow{\rho}F_{2}\xrightarrow{\phi}F_{1} \xrightarrow{\psi} R \rightarrow R/I \rightarrow 0\] with \(\mathrm{rank}\, F_{1} = \mathrm{rank}\, F_{2} = 3\), and \(\mathrm{rank}\, F_{1} = 1\). Finally, in the last section, the authors provide a complete description of the graded Betti sequence for those schemes which have \(n\) generators and syzygies with concentraded degrees with \(n\) is odd. Gorenstein ideals; graded Betti numbers; homological dimension; graded minimal free resolutions; structure theorems Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Configurations and arrangements of linear subspaces, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Minimal free resolutions for homogeneous ideals with Betti numbers \(1\), \(n\), \(n\), \(1\) | 0 |
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 note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems. Arzhantsev, Ivan; Flenner, Hubert; Kaliman, Shulim; Kutzschebauch, Frank; Zaidenberg, Mikhail: Flexible varieties and automorphism groups, Duke math. J. 162, No. 4, 767-823 (2013) Group actions on affine varieties, Group actions on varieties or schemes (quotients) Infinite transitivity on affine varieties | 0 |
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 Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang-Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Taalman, L., The Nash sheaf of a complete resolution, Manuscr. Math., 106, 249-270, (2001) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) The Nash sheaf of a complete resolution | 0 |
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 Given a linear group \(G\) over a field \(k\), we define a notion of index and residue of an element \(g\in G(k((t)))\). The index \(r(g)\) is a rational number and the residue a group homomorphism \(\operatorname{res}(g):\mathbb{G}_a\) or \(\mathbb{G}_m\to G\). This provides an alternative proof of Gabber's theorem stating that \(G\) has no subgroups isomorphic to \(\mathbb{G}_a\) or \(\mathbb{G}_m\) iff \(G(k[[t]])=G(k((t)))\). In the case of a reductive group, we offer an explicit connection with the theory of affine Grassmannians. Group schemes, Galois cohomology of linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Residues on affine Grassmannians | 0 |
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 cyclic group \(G\) acting on a smooth variety \(X\) with only one character occurring in the \(G\)-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold \([X / G]\) and the blow-up resolution \(\widetilde{Y} \rightarrow X / G\). Some results generalize known facts about \(X = \mathbb{A}^n\) with diagonal \(G\)-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals \(| G |\), we study the induced tensor products under the equivalence \(\mathrm{D}^{\mathrm{b}}(\widetilde{Y}) \cong \mathrm{D}^{\mathrm{b}}([X / G])\) and give a `flop-flop = twist' type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry Derived categories of resolutions of cyclic quotient singularities | 0 |
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 Deligne-Lusztig variety associated to minimal length elements in any \(\delta\)-conjugacy class of the Weyl group is affine, which was conjectured by \textit{S. Orlik} and \textit{M. Rapoport} [in J. Algebra 320, No. 3, 1220-1234 (2008; Zbl 1222.14102)].
Another proof of the same statement is given by \textit{C. Bonnafé} and \textit{R. Rouquier} [in J. Algebra 320, No. 3, 1200-1206 (2008; Zbl 1195.20048)]. Deligne-Lusztig varieties; finite groups of Lie type; Weyl groups; conjugacy classes; minimal length elements He, X., On the affineness of Deligne-Lusztig varieties, J. Algebra, 320, 3, 1207-1219, (2008) Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Linear algebraic groups over finite fields, Reflection and Coxeter groups (group-theoretic aspects) On the affineness of Deligne-Lusztig varieties. | 0 |
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 a smooth well-formed Picard rank-one Fano complete intersection of dimension at least 2 in a toric variety is a weighted complete intersection. Fano manifold; weighted complete intersection; toric variety Fano varieties, Complete intersections, Toric varieties, Newton polyhedra, Okounkov bodies Smooth prime Fano complete intersections in toric varieties | 0 |
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 \((X,0)\subset (\mathbb{C}^{n},0)\) be the germ of a complex variety at \(0\). We have two metrics on \((X,0)\): the outer metric \(d_{o}(x,y):=\left\vert\left\vert y-x\right\vert \right\vert \) and the inner metric \(d_{i}(x,y):=\inf \{\operatorname{length}(\phi ):\) \(\phi \) -- a path in \(X\) from \(x\) to \(y\},\) where length of \(\phi \) is in the Riemannian metric induced from \(\mathbb{C}^{n}\) on \(X.\) The main result is a necessary and sufficient condition for a normal surface singularity to have these metrics equivalent. The condition is the test of the equivalence of these metrics on a special class of pairs of arcs in \(X\) -- the nodal test curve criterion. The class of pairs of arcs is considerable reduced with respect to such a
class of pairs used by \textit{L. Birbrair} and \textit{R. Mendes} [in: Singularities and foliations. Geometry, topology and applications. BMMS 2/NBMS 3, Salvador, Brazil, 2015. Proceedings of the 3rd singularity theory meeting, ENSINO, July 8--11, 2015 and the Brazil-Mexico 2nd meeting of singularities, July 13--17, 2015. Cham: Springer. 549--553 (2018; Zbl 1405.14004)] to check the equivalence of these metrics of semi-algebraic sets. surface singularity; Lipschitz geometry; bilipschitz equivalence Singularities in algebraic geometry, Local complex singularities, Complex surface and hypersurface singularities A characterization of Lipschitz normally embedded surface singularities | 0 |
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 produces several examples of varieties \({\mathcal X}\) for which the global sections functor \(\Gamma({\mathcal X},\_): {\mathcal D}_{\mathcal X}\text{-\textbf{mod}}\to{\mathcal D}({\mathcal X})\text{-\textbf{mod}}\) is exact, and makes \({\mathcal D}({\mathcal X})\text{-\textbf{mod}}\) a quotient category of \({\mathcal D}_{\mathcal X}\text{-\textbf{mod}}\), but is not an equivalence. These varieties are quotients by finite group actions of \(\mathcal D\)-affine varieties. The torsion of \(\Gamma({\mathcal X},\_)\) is also described, in some cases. Here, \({\mathcal D}_{\mathcal X}\text{-\textbf{mod}}\) denotes the category of quasi-coherent \({\mathcal D}_{\mathcal X}\)-modules. global sections functor; quotient category; finite group actions; \(\mathcal D\)-affine varieties; category of quasi-coherent \({\mathcal D}_{\mathcal X}\)-modules Martin P. Holland, Varieties which are almost \?-affine, Bull. London Math. Soc. 25 (1993), no. 4, 321 -- 326. Rings of differential operators (associative algebraic aspects), Universal enveloping (super)algebras, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Module categories in associative algebras, Torsion theories; radicals on module categories (associative algebraic aspects) Varieties which are almost \({\mathcal D}\)-affine | 0 |
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 is essentially a very good mathematical-computational paper strongly emphasizing several important problems related to local mirror symmetry with special attention on the cases where a certain parameter \(k\) can be greater or equal to one. A straight forward calculational scheme is provided using the basic tools of equivariant \(I\)-functions and their Birkhoff factorization.
This article has 6 sections: Introduction, Overview (Background, Equivalent local mirror symmetry of curves, Non-nef toric varieties), Equivalent mirror symmetry (Equivalent Picard-Fuchs equation, Non-vanishing invariants, \(k\geq 1\), A model computation) Mirror symmetry (Motivation, Verification using \(I\)-function for \(k= 1, 2\), Alternative proof of connection matrices) Mirror symmetry for \(F_n\) and \(KF_n\), \(n\geq 3< KF_3\), connection matrices for \(F_3\), \(F_4\), Quantum differential equations) and Conclusion. This valuable contribution is of great current significance. local mirror symmetry; topological string; quantum cosmology Forbes, B.; Jinzenji, M., \(J\) functions, non-nef toric varieties and equivariant local mirror symmetry of curves. int, J. Mod. Phys. A, 22, 2327-2360, (2007) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies \(J\) functions, non-nef toric varieties and equivariant local mirror symmetry of curves | 0 |
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 crepant resolution conjecture (CRC) states that the orbifold quantum cohomology of an orbifold \(X/G\) is isomorphic to the (``ordinary'') quantum cohomology ring of a crepant resolution \(Y\to X/G\), up to a change of basis and a specialization of extra parameters on which the latter ring in general depends. The CRC is known to hold in many examples, including both local Calabi-Yau as well as Fano varieties.
Here, the CRC is verified for a simple, but interesting, compact Fano threefold: Let \(X\) be the flag variety of \({\mathbb C}^3\) equipped with the bilinear form \(\langle z,w\rangle=z_1w_3+z_2w_2+z_3w_1\), and \(\mathbb Z_2:X\to X\) be the involution sending a flag to its orthogonal complement. A crepant resolution \(Y\) of \(X/\mathbb Z_2\) is obtained by embedding \(X\) in \({\mathbb P}^2\times {\mathbb P}^2\), with involution exchanging factors, and blowing up the diagonal.
The proof utilizes both explicit calculation of low-degree Gromov-Witten invariants of \(X/\mathbb Z_2\) and \(Y\), as well as structural properties of the quantum cohomology rings (associativity, WDVV equations). Gromov-Witten; Orbifold Gillam, W. D., The crepant resolution conjecture for three-dimensional flags modulo an involution, Comm. Algebra, 41, 2, 736-764, (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles The crepant resolution conjecture for three-dimensional flags modulo an involution | 0 |
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 a survey dealing with the relation between Weil and Cartier divisors of a compact variety \(X^m\) on the one hand, and its \((2m-2)\)-th homology and second cohomology, respectively, on the other hand.
The main results are taken from two preprints written by the author and \textit{G. Barthel, K.-H. Fieseler}, and \textit{L. Kaup}: In case \(X\) is a compact toric variety, the canonical homomorphisms \(\text{Pic} (X)\to H^2(X)\) and \(\text{WeilDivCl} (X)\to H_{2m-2}(X)\) are isomorphisms. In particular, the natural inclusion \(\text{Pic} (X) \hookrightarrow \text{WeilDivCl} (X)\) corresponds to the Poincaré morphism.\ Moreover, the paper recalls the well known combinatorial description of all these groups for toric varieties.
At the end, the author mentions an interesting result obtained by the same authors concerning the intersection homology (middle perversity) of a toric variety. Sitting in between cohomology and homology, \(IH_{2m-2}(X)\) may be identified with those divisor classes on \(X\) that are represented by equivariant Weil divisors being Cartier in codimension two. Weil divisor; Cartier divisors; toric varieties; intersection homology Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, (Co)homology theory in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Poincaré morphism and divisors of compact toric varieties | 0 |
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 present paper is to report on a recent series of results that somewhat changes the conventional perspective (of resolution of singularities). Namely, upon closer inspection, it turns out that the holomorphic symplectic form, an auxiliary and almost accidental piece of structure on the resolution \(X\), actually insures all of the other good properties it enjoys -- the semismall property, the cohomological purity of the fibers, and so on and so forth. Moreover, the theory can be pushed through so far as to give a complete algebraic description of the derived category of coherent sheaves on the resolution \(X\). This gives new information even in the well-studied cases such as the
Springer resolution or the Hilbert scheme. quantization; Springer resolution; algebraic symplectic geometry; symplectic singularity D. Kaledin, Geometry and topology of symplectic resolutions , preprint,\arxivmath/0608143v1[math.AG] Global theory of symplectic and contact manifolds, Poisson manifolds; Poisson groupoids and algebroids, Global theory and resolution of singularities (algebro-geometric aspects) Geometry and topology of symplectic resolutions | 0 |
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 normal bundle of smooth Fano surface \(S\) inside a Calabi--Yau threefold \(X\) is a canonical bundle. It is negative, so a stable map to \(S\) can't be deformed to \(X\). Thus moduli spaces of stable maps to \(S\) and to \(X\) coincide. However their virtual fundamental classes are not; one for \(X\) is a cup-product of one for \(S\) with an Euler class of the so called obstruction bundle. Integrals of this Euler classes are called local Gromov--Witten invariants. They are defined in terms of Fano surface \(S\) and are equal to Gromov--Witten invariants of \(X\) if \(S\) can be embedded to a Calabi--Yau threefold \(X\).
If \(S\) is toric one can use localization technic to compute these invariants. In the paper authors define the generalization of local Gromov--Witten invariants for toric surfaces, not necessarily Fano. Using localization authors compute these invariants (Theorem 1). local Gromov--Witten invariants; localization; partition function; geometric engineering Yang, F.; Zhou, J.: Local Gromov--Witten invariants of canonical line bundles of toric surfaces, Sci. China math. 53, No. 6, 1571-1582 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Local Gromov-Witten invariants of canonical line bundles of toric surfaces | 0 |
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 paper is to give a survey about results on algebraic topology properties of toric varieties. Several books and papers give reviews of general results on toric varieties, we will concentrate on the algebraic topology point of view. More precisely, we will indicate how combinatoric definition of toric varieties provides interesting interpretation in ordinary homology and intersection homology: Poincaré morphism, Betti numbers, divisors, decomposition theorem and Gorenstein singularities. toric varieties; intersection homology; Poincaré morphism; Betti numbers; divisors; decomposition; Gorenstein singularities Toric varieties, Newton polyhedra, Okounkov bodies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Singularities in algebraic geometry Algebraic topology and singularities of toric varieties | 0 |
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 such that its derived category of coherent sheaves, \(D^b(X)\) admits a \textit{full strongly exceptional collection} \((U^a)_{a\in A}\) (cf. [\textit{D. Huybrechts}, Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. Oxford Science Publications. Oxford: Clarendon Press (2006; Zbl 1095.14002)]). Then every coherent sheaf \(\mathscr F\) can be represented by a complex whose term are direct sums of elements in \((U^a)\). This is called \textit{Beilison monad of \(\mathscr F\)}. A \textit{Tate resolution} for \(\mathcal F\) is a a doubly infinite complex which simultaneously packages all the cohomology spaces and the Beilison monads of all the twists of \(\mathscr F\). The construction of a Tate resolution for sheaves on the projective space yields in many cases the fastest algorithms for computing cohomology (cf. [\textit{D. Eisenbud} et al., Trans. Am. Math. Soc. 355, No. 11, 4397--4426 (2003; Zbl 1063.14021)]).
In this paper, masterly written, the authors show the existence of and provide algorithms to construct Tate resolutions for sheaves on the product of the projective spaces. As application to their work the prove a splitting criterion for vector bundles over products of projective spaces which broadly generalize Horrocks' criterion. Tate resolution; BGG correspondence; free resolution; cohomology computing algorithm Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Effectivity, complexity and computational aspects of algebraic geometry, Syzygies, resolutions, complexes in associative algebras Tate resolutions for products of projective spaces | 0 |
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 scheme defined over an algebraically closed field \(k\) of characteristic \(p\) and let \(\mathcal D_X\) be the sheaf of \(k\)-linear differential operators on \(X\). The scheme \(X\) is called \(D\)-affine if every \(\mathcal D_X\)-module \(M\) is generated over \(\mathcal D_X\) by its global sections and \(H^i(X,M) = 0\) for all \(i > 0\). First, the author recalls some well known facts. For example, every flag variety in characteristic zero is \(D\)-affine (see [\textit{A. Beilinson} and \textit{J. Bernstein}, C. R. Acad. Sci., Paris, Sér. I 292, 15--18 (1981; Zbl 0476.14019)]). However, in positive characteristic it is not true (see [\textit{M. Kashiwara} and \textit{N. Lauritzen}, C. R., Math., Acad. Sci. Paris 335, No. 12, 993--996 (2002; Zbl 1016.14009)]), although some flag varieties are still \(D\)-affine. Next, any smooth \(D\)-affine projective toric variety is a product of projective spaces (see [\textit{J. F. Thomsen}, Bull. Lond. Math. Soc. 29, No. 3, 317--321 (1997; Zbl 0881.14020)]), etc. In the paper under review the author describes some further results concerning the classification of smooth projective \(D\)-affine varieties over fields of any characteristic. In particular, he proves that any smooth projective \(D\)-affine variety is algebraically simply connected and its image under a fibration is \(D\)-affine as well. In zero characteristic such \(D\)-affine varieties are, in fact, uniruled. Moreover, assuming \(p=0\) or \(p>7\), he shows that a smooth projective surface is \(D\)-affine if and only if it is isomorphic to either \(\mathbb P^2\) or \(\mathbb P^1\times \mathbb P^1\). Some results are also obtained for three-folds \(D\)-affine varieties. In positive characteristic case the author applies his own modified version of the generic semipositivity theorem due to \textit{Y. Miyaoka} [Proc. Symp. Pure Math. 46, No. 1, 245--268 (1987; Zbl 0659.14008)]. smooth projective varieties; flag varieties; Miyaoka's semipositivity theorem; cotangent bundle; rational surfaces; divisorial contractions; fibrations; crystalline differential operators; étale fundamental group; semistability; reflexives sheaves; semipositive sheaves; uniruled varieties; Riemann-Hilbert correspondence; stable Higgs bundle; Chern classes; flat connections; Artin's criterion of contractibility; Kodaira dimension; Hirzebruch surface; canonical divisor; surfaces of general type; Barlow's surfaces; del Pezzo surfaces; Fano three-folds Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules On smooth projective D-affine varieties | 0 |
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 investigate regular maps from real algebraic varieties into real Fermat varieties. It is proved that under some natural assumptions, all such maps are null homotopic. Kucharz, W., Regular maps into real Fermat varieties, Bull. Lond. Math. Soc., 451, 1086-1090, (2013) Real algebraic sets, Topology of real algebraic varieties Regular maps into real Fermat varieties | 0 |
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 Extending work of \textit{R. Bielawski} and \textit{A. S. Dancer} [Commun. Anal. Geom. 8, 727-760 (2000; Zbl 0992.53034)] and \textit{H. Konno} [Int. J. Math. 11, 1001-1026 (2000; Zbl 0991.53027)], we develop a theory of toric hyper-Kähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyper-Kähler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the hard Lefschetz theorem and the volume polynomials of Khovanskij-Pukhlikov [\textit{A. G. Khovanskij} and \textit{A. V. Pukhlikov}, St. Petersbg. Math. J. 4, 789-812 (1993; Zbl 0798.52010)], are extended to the hyper-Kähler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of \textit{H. Nakajima} [Duke Math. J. 91, 515-560 (1998; Zbl 0970.17017)]. affine spaces; quotients; torus actions; matroid; toric quiver varieties T. Hausel and B. Sturmfels, \textit{Toric hyper-Kähler varieties}, \textit{Doc. Math.}\textbf{7} (2002) 495 [math/0203096]. Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Toric varieties, Newton polyhedra, Okounkov bodies Toric hyperkähler varieties | 0 |
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 \(\mathbb{Q}\)-Gorenstein variety. If \(K_X\) is a \(N\)-th multiple (with \(N\in\mathbb{Q}_{>0}\))
inside \(\text{Pic} X\), then it follows from the author's paper [Tohoku Math. J., II. Ser. 55, No. 4, 551--564 (2003; Zbl 1078.14077)] that \(N\leq n\) unless \(X=\mathbb{P}^n\). As a sequel to [loc. cit.], the present paper is devoted to the next extremal case, namely to \(N=n\). If \(X\) is \(\mathbb{Q}\)-factorial, then this equality implies that either
\(X=\mathbb{P}_{\mathbb{P}^1}(\oplus_i\mathcal O(q_i))\) is a projective bundle or that \(X=\mathbb{P}(1,1,2,\ldots,2)\).
In the non-\(\mathbb{Q}\)-factorial case, \(X\) equals the target space of the flopping contraction of \(\mathbb{P}_{\mathbb{P}^1}(\mathcal O^{n-2}\oplus\mathcal O(1)^2)\). O. Fujino, Toric varieties whose canonical divisors are divisible by their dimensions, Osaka J. Math. 43 (2006), 275-281. Toric varieties, Newton polyhedra, Okounkov bodies, Minimal model program (Mori theory, extremal rays) Toric varieties whose canonical divisors are divisible by their dimensions | 0 |
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 \({\mathcal A}_{g,T}\) be the moduli space of complex abelian varieties of dimension \(g\) with a polarization of type \(T = (t_1, \dots, t_g)\), where \(t_1, \dots, t_g\) are positive integers such that \(t_1 |\cdots |t_g\). The purpose of this paper is to prove that the quasi-projective variety \({\mathcal A}_{g,T}\) is of general type for \(g \geq 16\), and for \(g \geq 8\) if each \(t_i\) is odd and a sum of two squares (it has been proved by \textit{Freitag}, \textit{Tai} and \textit{Mumford} that \({\mathcal A}_{g, (1, \dots, 1)}\) is of general type for \(g \geq 7)\). The variety \({\mathcal A}_{g,T}\) can be constructed as the quotient of the Siegel upper half-space \(S_g\) by a certain subgroup \(\Gamma_T\) of \(Sp (2g,\mathbb{Q})\). One needs to construct enough \(k\)-canonical forms on a smooth projective model of \({\mathcal A}_{g,T}\) and for that, by now standard techniques, it is enough to construct one non-zero \(\Gamma_T\)-modular form on \(S_g\) of weight \(k(g + 1)\), that vanishes to order \(> k\) at infinity.
This is done by considering an embedding \(h\) of \(S_g\) into the quaternionic half-space \(S_{g,H}\), equivariant for the respective actions of \(\Gamma_T \) and of a certain subgroup \(\Gamma_{q,H}\) of \(GL(2g, {\mathfrak O})\) (where \({\mathfrak O}\) is the ring of Hurwitz quaternions), independent of \(T\), such that for each \(\Gamma_{g,H}\)-modular form \(F\) on \(S_{g,H}\), the composition \(F \circ h\) is a \(\Gamma_T\)-modular form of the same weight, that vanishes to the same order at infinity. Using analogs of theta constants, one constructs a non-zero \(\Gamma_{g,H}\)-modular form \(F\) of weight \(2^{2g + 5} n\) (for some integer \(n)\) that vanishes to order at least \(2^{2g + 1} n\) at infinity. For \(g \geq 16\), one has \(2^{2g + 1} n (g + 1) > 2^{2g + 5} n\), hence \((F \circ h)^{g + 1}\) is the required modular form.
When each \(t_i\) is odd and a sum of two squares, the technique is similar: one considers an equivariant embedding \(h\) of \(S_g\) into the Hermitian half-space \(S_{g,C}\), and one constructs a non-zero \(\Gamma_{g,C}\)-modular form \(F\) of weight \(2^{g + 2} (2^g + 1) (g + 1)\) that vanishes to order at least \(2^{2g - 1} (g + 1)\) at infinity. For \(g \geq 8\), one has \(2^{2g - 1} (g + 1) > 2^{g + 2} (2^g + 1)\), hence \(F \circ h\) is the required modular form. moduli space of complex abelian varieties; polarization; quotient of the Siegel upper half-space; theta constants; modular form Tai, Y.-S.: On the Kodaira dimension of moduli spaces of Abelian varieties with non-principal polarizations. Abelian varieties (Egloffstein, 1993) (Berlin), pp. 293-302. Berlin: de Gruyter 1995 Algebraic moduli of abelian varieties, classification, Theta functions and abelian varieties, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms On the Kodaira dimension of moduli spaces of abelian varieties with non-principal polarizations | 0 |
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 determine the minimal free resolutions of some configurations of lines in a projective space \(\mathbb P^{n}_{K}\) over an infinite field \(K\): the complete grids of lines (the projective closure of affine lines that are parallel to the coordinate axes and pass through a lattice of points), and the complete pseudo-grids. They determine the total Betti numbers of \(m\)-fat complete grids and pseudo-grids. The authors prove that complete pseudo-grids are seminormal under certain conditions, contrasting them with complete grids. Quoting from the authors' abstract: `The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals.' and `we give new contributions, in particular about the maps of the resolution.' complete grid of lines; lifting and pseudo-lifting; mapping cone; resolution; seminormality Syzygies, resolutions, complexes and commutative rings, Computational homological algebra, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) From grids to pseudo-grids of lines: resolution and seminormality | 0 |
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 \({\mathcal L}\) be a distributive lattice equipped with lattice operations \(\wedge\) and \(\vee\), \(n=\sharp({\mathcal L})\) and \({\mathbb{A}}^n\) be the affine \(n\)-dimensional space \(\mathrm{Spec} (K[X_\alpha, \alpha\in {\mathcal L}])\) where \(K\) is any algebraically closed field of arbitary characteristic. Let \(X_{\mathcal L}\) be the affine subvariety of \({\mathbb{A}}^n\) defined by the ideal generated by binomials of the form
\[
X_\tau X_\phi-X_{\tau\vee\phi}X_{\tau\wedge \phi}.
\]
It turns out that \(X_{\mathcal L}\) is a normal toric variety, called \textsl{Hibi toric variety}. The main achievement of the paper under review is the explicit construction of bases for the cotangent spaces (Section 5) at the singular points of Hibi toric varieties, i.e. toric varieties which are associated to finite distributive lattices. The singular loci of Hibi toric varieties \(X_{\mathcal L}\) are determined (Theorem~6.9). distributive lattices; Hibi toric varieties; Schubert varieties; minuscule homogeneous varieties; singular loci of Hibi varieties Lakshmibai, V. and Mukherjee, H., Singular loci of {H}ibi toric varieties, Journal of the Ramanujan Mathematical Society, 26, 1, 1-29, (2011) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Singular loci of Hibi toric varieties | 0 |
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 any ample line bundle \(L\) on a projective toric variety of dimension \(n\), it is proved that the line bundle \(L^{\otimes i}\) is normally generated if \(i\) is greater than or equal to \(n-1\), and examples showing that this estimate is best possible are given. Moreover we prove an estimate for the degree of the generators of the ideals defining projective toric varieties. In particular, when \(L\) is normally generated, the defining ideal of the variety embedded by the global sections of \(L\) has generators of degree at most \(n+1\). When the variety is embedded by the global sections of \(L^{\otimes(n - 1)}\), then the defining ideal has generators of degree at most three. toric variety; generators of defining ideal; ample line bundle Ogata, S.; Nakagawa, K., On generators of ideals defining projective toric varieties, Manuscripta Mathematica, 108, 33-42, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On generators of ideals defining projective toric varieties | 0 |
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 Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of \textit{toric co-Higgs bundle}. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Toric varieties, Newton polyhedra, Okounkov bodies, Holomorphic bundles and generalizations Toric co-Higgs bundles on toric varieties | 0 |
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 introduce the notion of strong regular embeddings of Deligne-Mumford stacks. These morphisms naturally arise in the related contexts of generalized Euler sequences and hypertoric geometry. Edidin, D., Strong regular embeddings of Deligne-Mumford stacks and hypertoric geometry, Michigan math. J., 65, 2, 389-412, (2016), arXiv:150304828 Stacks and moduli problems, Generalizations (algebraic spaces, stacks), Toric varieties, Newton polyhedra, Okounkov bodies, (Equivariant) Chow groups and rings; motives Strong regular embeddings of Deligne-Mumford stacks and hypertoric geometry | 0 |
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 codimension \(1\) subvariety of dimension greater than \(1\) of a variety of minimal degree \(Y.\) If \(X\) is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then \(X\) is arithmetically Gorenstein and the authors characterize such subvarieties \(X\) of \(Y\), via apolarity, as those whose apolar hypersurfaces are Fermat.
In this paper, the authors extensively use the concept of an arithmetically Cohen-Macaulay projective variety (aCM for short), and the concept of an arithmetically Gorenstein variety (aG for short).
They extend the results of \textit{M. L. Green} [Duke Math. J. 49, 1087--1113 (1982; Zbl 0607.14005)] with the use of the general Kodaira vanishing theorem. They consider normal projective \(n\)-dimensional aCM varieties \(X \subset \mathbb{P}^N\) with canonical Gorenstein singularities that are regular (i.e. \(h^{1,0}(X) = 0\)) for which \(\omega_X\) is base-point free, the image of the canonical map \(\phi_{\omega_X}\) has maximal dimension and \(h^0(X, \omega_X) \geq n + 2\). They show that, for such varieties, the canonical ring \(R\) of \(X\) is generated in degree \(n\) unless the image of the canonical map is a variety of minimal degree, in which case \(R\) is generated by elements of degree at most \(n + 1\). The above results are also generalized to \(s\)-subcanonical varieties; see Proposition 1.
We also give conditions under which a projective variety \(X \subset \mathbb{P}^N\) with canonical Gorenstein singularities is aG. We show (See Theorem 6 and Theorem 8) that this happens if \(X\) is s-subcanonical and \(\ell\)-normal, for all \(\ell\) with \(0 \leq \ell \leq n + s -1\) and, if \(s \geq 0\), satisfies the additional condition that \(h^i(X,O_X(k)) = 0\) for \(1 \leq i \leq n-1\) and \(0 \leq k \leq s\).
In this paper, the authors also prove natural generalizations of the celebrated Noether and Enriques-Petri-Babbage theorems in the wider context of s- subcanonically regular varieties (See Theorem 23).
The main theorem is the first step to study the geometry of an \(s\)- subcanonically regular variety of dimension \(n\) via the behaviour of the rational map \(\alpha_X : G(m,N) \rightarrow H_{m,s+n+1}\). For a non-trivial example concerning the canonical curve case see \textit{E. Ballico, G. Casnati} and \textit{R. Notari} [J. Algebra 332, No. 1, 229--243 (2011; Zbl 1242.14030)].
The assumption that the resolution is crepant establishes an interesting link between the theory of singularities and the theory of apolarity. Moreover, some of the geometry described in this paper could shed some light on some aspects of Artinian Gorenstein Rings, see [\textit{A. Conca, M. E. Rossi} and \textit{G. Valla}, Compos. Math. 129, No. 1, 95--121 (2001; Zbl 1030.13005)]. Gorenstein variety; Subcanonical varieties; apolarity \(n\)-folds (\(n>4\)), Fano varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Projective techniques in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On subcanonical Gorenstein varieties and apolarity | 0 |
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 studies minimal free resolutions of ideal sheaves of projective curves \(C \subset {\mathbb P}^n\). Let
\[
0 \rightarrow E_n \rightarrow\dots\rightarrow E_1 \rightarrow {\mathcal I}_C \rightarrow 0
\]
be a minimal resolution of the ideal sheaf of \(C\); we say that \(C\) has property \(N_0\) if it is a.C.M. and that it has property \(N_p\), \(p\geq 1\), if \(N_0\) holds and each \(E_i \equiv \sum {\mathcal O}_{{\mathbb P}^n}(-i-1)\). In a previous paper [Pac. J. Math. 172, No. 2, 315-319 (1996; Zbl 0848.14007)], \textit{E. Ballico} has given a bound \(G(p,n)\) such that for each \(g \leq G(p,n)\), the general linearly normal non-special curve \(C \subset {\mathbb P}^n\), with genus \(g\) and degree \(g+n\) satisfies property \(N_p\). The numbers are such that we get property \(N_p\) for \(\deg(C)g + \sqrt {2(p+1)g}\). In this paper the methods used to get the previous result are generalized in order to get the same statement also for \(k\)-gonal curves, where \(k > n \geq p\), \(n \geq 3\). Moreover, results are given about the linearity of the resolution also in the case of non-linearly normal, non-special projective curves. minimal free resolutions of ideal sheaves; special divisors; projective curves; \(k\)-gonal curves; linearity of the resolution Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves On the minimal free resolution of general \(k\)-gonal curves | 0 |
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 geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to \(2n-2\), is exactly \(2n-2\) if and only if the cone is a bipyramidal cone, where \(n>1\) is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones. Michelacakis, N. J.; Thoma, A.: On the geometry of complete intersection toric varieties. Arch. math. 87, 113-123 (2006) Toric varieties, Newton polyhedra, Okounkov bodies, Complete intersections, \(n\)-dimensional polytopes On the geometry of complete intersection toric varieties | 0 |
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 compact torus with Lie algebra \(\mathfrak g\) and integral lattice \(\mathbb Z_G\subset \mathfrak g\). Let \(u_1,\dots,u_N\in\mathbb Z_G\) be a set of primitive vectors generating the \(\mathbb R\)-vector space \(\mathfrak g\), and for \(\lambda_1,\dots\lambda_N\in \mathbb R\) consider the polyhedral set \(P=P_{u,\lambda}:=\{\eta\in\mathfrak g^*\,| \,\langle\eta,u_j\rangle -\lambda_j\geq 0, 1\leq j\leq N\}\) when \(P\) has nonempty interior and the collection of inequalities defining \(P\) is minimal.
The authors modify a construction of \textit{T. Delzant} [Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)] who established a one-to-one correspondence between the class of \(2n\)-dimensional compact toric symplectic manifolds and a specific class of convex polytopes in \(\mathbb R^n\), now known as Delzant polytopes. The Delzant polytopes are the images of the momentum maps of those manifolds.
In the present paper the authors associate to \(P\) a symplectic stratified space \(M\) with an effective Hamiltonian action of \(G\) and momentum map \(\phi:M\rightarrow\mathfrak g^*\) with \(\phi(M)=P\). The space \(M\) is a symplectic quotient of \(\mathbb C^N\) by a compact subgroup of the standard torus \(T^N\) and therefore a Kählerian complex space. Moreover \(M\) is a toric space, since the \(G\)-action on \(M\) extends to an analytic action of the complexification \(G^{\mathbb C}\) of \(G\) with dense open orbit. For proofs of these facts see the results of \textit{P. Heinzner, A. Huckleberry} and \textit{F. Loose} in [Geom. Funct. Anal. 4, No. 3, 288--297 (1994; Zbl 0816.53018), J. Reine Angew. Math. 455, 123--140 (1994; Zbl 0803.53042), Invent. Math. 126, No. 1, 65--84 (1996; Zbl 0855.58025)]. The \(\phi\)-preimages of the open faces of \(P\) are also quotients of this type and submanifolds of \(M\).
The authors give explicit formulas for Kähler potentials on these manifolds, hereby generalising a result of V. Guillemin for compact \(M\): the function \(\sum_{j=1}^N \lambda_j\log(u_j-\lambda_j)+u_j)\circ\phi\) is a Kähler potential on the \(\phi\)-preimage of the interior of \(P\), where \(u_j\) is considered as a function on \(\mathfrak g^*\).
For related results see also \textit{D. M. Calderbank, L. David} and \textit{P. Gauduchon} [J. Symplectic Geom. 1, No. 4, 767--784 (2003; Zbl 1155.53336)] and \textit{D. Burns} and \textit{V. Guillemin} [Commun. Anal. Geom. 12, No. 1--2, 281--303 (2004; Zbl 1073.53113)]. Kähler potential; symplectic quotient; Kähler toric space; momentum map; Delzant polytope Burns, D; Guillemin, V; Lerman, E, Kähler metrics on singular toric varieties, Pac. J. Math., 238, 27-40, (2008) Compact Kähler manifolds: generalizations, classification, Kähler manifolds, Momentum maps; symplectic reduction, Toric varieties, Newton polyhedra, Okounkov bodies Kähler metrics on singular toric varieties | 0 |
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 studies effective very ampleness and projective normality of line bundles of certain Calabi-Yau varieties and hyperhähler varieties. The first main result states that for an ample line bunde \(A\) on a manifold with trivial canonical bundle of dimension \(4\), \(A^{\otimes n}\) is very ample and embeds \(X\) as a projectively normal variety for \(n\geq 16\) (for \(n\geq 15\) if furthermore \(h^1(\mathcal{O}_X)=0\)). The second main result studies projective normality for ample and globally generated line bundles on known examples of hyperhähler manifolds. Such results generalize known results of line bundles of \(K3\) surfaces to higher dimensions. Calabi-Yau varieties; hyperkähler varieties; projective normality Calabi-Yau manifolds (algebro-geometric aspects), Holomorphic symplectic varieties, hyper-Kähler varieties, Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Remarks on projective normality for certain Calabi-Yau and hyperkähler varieties | 0 |
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 algebraic toric variety with respect to an action of an algebraic torus \(S\). Let \(\Sigma\) be the corresponding fan. The aim of this paper is to investigate open subsets of \(X\) with a good quotient by the (induced) action of a subtorus \(T\subset S\). It turns out that it is enough to consider open \(S\)-invariant subsets of \(X\) with a good quotient by \(T\). These subsets can be described by subfans of \(\Sigma\). We give a description of such subfans and also a description of fans corresponding to quotient varieties. Moreover, we give conditions for a subfan to define an open subset with a complete quotient space. quotients of toric varieties; actions of subtori; subfans; complete quotient space Świe\ogonek cicka, J.: Quotients of toric varieties by actions of subtori. Colloq. math. 82, No. 1, 105-116 (1999) Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Quotients of toric varieties by actions of subtori | 0 |
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 studies submanifolds of real algebraic sets. The main problem is to determine when a submanifold can be slightly moved to become an algebraic subset. The authors give a good and quick review of the necessary background material from algebraic geometry. The work relies heavily upon the techniques of differential topology.
The main result is to show that a codimension-1 smooth submanifold of a nonsingular algebraic set is isotopic to a nonsingular algebraic subset when a certain obstruction vanishes. Related results and examples of a similar nature are also included. blowing up; submanifolds of real algebraic sets; algebraic subset; isotopic S. Akbulut and H. King, Submanifolds and homology of nonsingular real algebraic varieties , Amer. J. Math. 107 (1985), no. 1, 45-83. JSTOR: Neighborhoods of submanifolds, Isotopy and pseudo-isotopy, Real algebraic and real-analytic geometry, Birational geometry Submanifolds and homology of nonsingular real algebraic varieties | 0 |
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 \(({\mathcal C},0)\) be an irreducible germ of a complex plane curve. Let \(R\) be the local analytic algebra of \({\mathcal C}\) and \(\nu\) be the unique valuation of \(R\). The graded ring gr\(_{\nu}(R):= \bigoplus_n {I_n\over I_{n+1}}\) where \(I_n:=\{x \in R|\nu(x)\geq n\}\), \(n\in \mathbb{N}\), is the ring of the monomial curve \({\mathcal C}^{\Gamma}\) with the same semigroup \(\Gamma\) as \({\mathcal C}\). There exists a one parameter deformation of \({\mathcal C}^{\Gamma}\) having all its fibers except the special one isomorphic to \({\mathcal C}\). In this paper, the authors show that \({\mathcal C}^{\Gamma}\) can be resolved by a single toric modification of its ambient space \(\mathbb C ^{g+1}\) (\(g\) is the number of Puiseux exponents of \({\mathcal C}\)). Furthermore, \({\mathcal C}\) may be naturally embedded in \(\mathbb C ^{g+1}\) in such a way that the toric modification above resolves \textit{both} \({\mathcal C}\) and \({\mathcal C}^{\Gamma}\).
This result leads to a difficult question: Given any germ of an analytic space and a valuation of its local algebra, can the germ be embedded in an affine space in such a way that a toric modification of the ambient space can resolve the singularity at the center of the valuation?
In his thesis (2002), \textit{Pedro Gonzalez Perez} proved that, in the same way, an irreducible quasi-ordinary hypersurface singularity can be resolved with one toric modification of a suitable ambient space where the quasi-ordinary singularity is a deformation of a toric singularity. In the general case, the graded ring gr\(_{\nu}(R)\) is not always Noetherian and the structure of the semigroup \(\Gamma\) can be very complicated. So the authors' question is far from being easy. resolving singularities; germ of complex plane curves; desingularization; valuation; toric modification; deformation of a toric singularity R. Goldin and B. Teissier, Resolving singularities of plane analytic branches with one toric morphism, Resolution of Singularities, A research textbook in tribute to Oscar Zariski, (eds. H. Hauser, J. Lipman, F. Oort and A. Quiros), Progr. Math., 181 , Birkhäuser, Basel, 2000, pp.,315-340. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Resolving singularities of plane analytic branches with one toric morphism | 0 |
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 how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on \(D3\) and \(D7\)-branes on (the partially resolved) \((dP_0)^3\) singularity, its embedding in a specific Calabi-Yau manifold as a hypersurface in a toric variety, the related type IIB orientifold compactification, as well as the corresponding F-theory uplift. Our techniques generalize naturally to complete intersections, and to a large class of F-theory backgrounds with singularities. compactification and string models; D-branes; F-theory; superstring vacua J. Ecker, \textit{Type IIA string theory on T}\^{}\{6\}\(/\)(\(Z\)\_{}\{2\} {\(\times\)} \(Z\)\_{}\{6\} {\(\times\)} {\(\Omega\)}\(R\))\textit{: model building and string phenomenology with intersecting D}6\textit{-branes}, Ph.D. thesis, Mainz U., Mainz Germany, (2016) [INSPIRE]. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Calabi-Yau theory (complex-analytic aspects) Global embeddings for branes at toric singularities | 0 |
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 considers complex analytic varieties whose singularities can be resolved by blow-ups along smooth centers, intersections of smooth normal crossing divisors. An explicit (based on matrices) algorithm is given in order to replace the finite sequence of blow-ups by just one blow-up along a suitable coherent sheaf of ideals locally generated by monomials in some system of coordinates. resolving singularities by blow-ups (C.) ( Grant Melles ) & (P.) ( Milman ) .- Single-Step Combinatorial Resolution via Coherent Sheaves of Ideals , Singularities in Algebraic and Analytic Geometry, Contemporary Mathematics , American Mathematical Society , 2000 , p. 77 - 88 MR 1792150 | Zbl 0973.14007 Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Single-step combinatorial resolution via coherent sheaves of ideals | 0 |
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, let A be an abelian variety defined over k and let L be an ample line bundle on A.- Our purpose is to give a condition for the normal generation of \(L^{\otimes 2}\). The result is as follows:
Theorem. If char(k)\(\neq 2\) and L is a symmetric ample line bundle, then \(L^{\otimes 2}\) is normally generated if and only if the origin of A is not contained in \(Bs| L\otimes P_{\alpha}|\) for any \(\alpha \in \hat A_ 2=\{\alpha \in \hat A; 2\alpha =0\}\) where \(\hat A\) is the dual abelian variety of A, P is the Poincaré bundle on \(A\times \hat A\), \(P_{\alpha}=P_{A\times \{\alpha \}}\) for \(\alpha\in \hat A\) and \(Bs| L\otimes P_{\alpha}|\) is the set of all base points of \(L\otimes P_{\alpha}\). normal generation of ample line bundles; abelian variety Akira Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 4, 119 -- 120. Akira Ohbuchi, A note on the projective normality of special line bundles on abelian varieties, Tsukuba J. Math. 12 (1988), no. 2, 341 -- 352. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic theory of abelian varieties, Divisors, linear systems, invertible sheaves A note on the normal generation of ample line bundles on abelian varieties | 0 |
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 develop a theory of \textit{irrational toric varieties} associated to arbitrary fans in real vector spaces, and use this theory to prove that the moduli space of Hausdorff limits of a projective irrational toric variety with the secondary polytope of a configuration of real vectors \(\mathcal{A}\).
The results in this paper parallel some on classical toric varieties but their proofs require different methods as fundamental facts from algebraic geometry do not hold for irrational toric varieties. These methods are condensed from [\textit{A. Pir}, Irrational toric varieties. Texas A\& M University (PhD Thesis) (2018), \url{https://core.ac.uk/download/pdf/187124453.pdf}].
The main motivation to study toric varieties comes from applications of toric varieties in geometric modeling, see for instance \textit{R. Krasauskas}, Adv. Comput. Math. 17, No. 1--2, 89--113 (2002; Zbl 0997.65027); \textit{G. Craciun} et al., Lect. Notes Comput. Sci. 5862, 111--135 (2010; Zbl 1274.65033); \textit{L. García-Puente} et al., ACM Trans. Graphics, 30(5), \#110 (2011)]. In \textit{E. Postinghel} et al., J. Lond. Math. Soc., II. Ser. 92, No. 2, 223--241 (2015; Zbl 1366.14047)] it was proved that when \(\mathcal{A}\) is a configuration of real vectors, the space of translates of the irrational toric variety \(Z_{\mathcal{A}}\) and their Hausdorff limits can be understood set-theoretically in terms of the secondary fan of \(\mathcal{A}\). In this paper, the authors promote this set-theoretical approach. secondary polytope; irrational toric variety; moment map; log-linear model; LVM manifold Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic and real-analytic geometry Irrational toric varieties and secondary polytopes | 0 |
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 Recall [\textit{M. Goresky} and \textit{R. MacPherson}, Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)] that a resolution \(p:\widetilde{X}\to X\) of an irreducible complex variety \(X\) is said to be small if, for each \(i>0\), one has \(\text{codim}_X \{x\in X\mid\dim p^{-1}(x)\geq i\}>2i\).
Let \(G=Sp(2n,\mathbb{C})\) or \(SO(2n,\mathbb{C})\), and let \(P=\mathbb{P}_n\). Recall [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1-54 (1978; Zbl 0447.14011)] that the Schubert varieties in \(Sp(2n,\mathbb{C})/P\) are indexed by \(\bigcup_{0\leq r\leq n} I_{n,r}\) where \(I_{n,r}=\{(\lambda_,\dots,\lambda_r)\mid 1\leq\lambda_1<\cdots<\lambda_r\leq n\}\).
Similarly, the Schubert varieties in \(SO(2n)/\mathbb{P}_n\) are labelled by the set \(\bigcup_{\substack{ 0\leq r\leq n\\ (n-r)\text{ even}}} I_{n,r}\).
The main results of this paper are:
Theorem 1.1. Let \(\lambda=(\lambda_1,\dots,\lambda_r)\in I_{n,r}\).
(i) The Schubert variety \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) has a small resolution if \(\lambda_r\leq n-r\).
(ii) Assume \(n-r\) is even so that \(\lambda\) gives rise to a Schubert variety \(X(\lambda)\) in \(SO(2n)/\mathbb{P}_n\). \(X(\lambda)\) has a small resolution if: (a) \(\lambda_r<n-r\); or (b) for \(r\geq 2\), \(\lambda_r=n\), \(\lambda_{r-1}\leq n-r\).
Theorem 1.2. Let \(\lambda=(n)\), \(n\geq 3\), and let \(Q\) be any parabolic subgroup contained in \(\mathbb{P}_n\subset Sp(2n)\). Let \(X(\Lambda)\) be the inverse image of \(X(\lambda)\subset Sp(2n)/\mathbb{P}_n\) under the projection \(Sp(2n)/Q\to Sp(2n)/\mathbb{P}_n\). Then \(X(\Lambda)\) does not admit any small resolution. Schubert variety; small resolution Sankaran P., Vanchinathan P.: Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians. Publ. Res. Inst. Math. Sci. 30(3), 443--458 (1994) Low codimension problems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians | 0 |
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 \(R\) be a Noetherian ring of characteristic \(p > 0\) of dimension \(d\) and \(I \subseteq R\) be an ideal of finite colength. Let \(M\) be a finitely generated \(R\)-module. The Hilbert-Kunz multiplicity of \(M\) with respect to \(I\) is defined as
\[
e_{H \, K}(M,I) = \lim_{n\rightarrow \infty} \frac{\ell(M/I^{[q]}M)}{q^{d}},
\]
where \(q = p^{n}\), \(I^{[q]}\) is the \(n\)-th Frobenius power of \(I\), and \(\ell(M/I^{[q]}M)\) denotes the length of the \(R\)-module \(M/I^{[q]}M\). The Hilbert-Kunz density function is defined as a compactly supported continuous function \(\mathrm{HKd}(M,I) : [0, \infty) \rightarrow [0, \infty)\) such that
\[
e_{H \, K}(M,I) = \int_{0}^{\infty}\mathrm{HKd}(M,I)(x) dx.
\]
In the paper under review, the authors study Hilbert-Kunz density functions and the asymptotic Hilbert-Kunz multiplicity for toric pairs \((X,D)\), where \(X\) is a projective toric variety of dimension \(d-1 \geq 1\) over an algebraically closed field \(K\) of characteristic \(p>0\) and \(D\) is a very ample torus invariant Cartier divisor. It is well-known that if \((X,D)\) is a toric pair, then we have an associated lattice polytope \(P_{D} \subset \mathbb{R}^{d-1}\). The first result of the paper gives an interpretation of the Hilbert-Kunz density function for a toric pair \((X,D)\) in terms of \(\mathcal{P}_{D}\).
Theorem A. Let us denote by \(C_{D}\) the convex rational polyhedral cone spanned by \(P_{D} \times 1\) in \(\mathbb{R}^{d}\). Let
\[
\mathcal{P}_{D} = \{p \in C_{d} \, | \, p \not \in (u_{j}, 1) + C_{d} \text{ for every } u_{j} \in P_{D} \cap \mathbb{Z}^{d-1}\}.
\]
Then the Hilbert-Kunz density function \(\mathrm{HKd}(X,D)\) for a toric pair \((X,D)\) is given by the sectional volume of the closure of \(\overline{\mathcal{P}_{D}}\), i.e.,
\[
\mathrm{HKd}(X,D)(\lambda) = \mathrm{Vol}_{d-1}(\overline{\mathcal{P}_{D}} \cap \{z = \lambda\})
\]
for some \(\lambda\geq 0\).
In order to get the main result of the paper, we also need the following (technical) proposition.
Proposition. Let \((X,D)\) be a toric pair, then for \(\lambda \geq 0\) we have
\[
\mathrm{HKd}(X,kD)(\lambda +1) = \frac{e_{0}(X,D)k^{d-1}}{(d-1)!} \phi_{kD}(\lambda) + O(k^{d-2}),
\]
where \(\phi_{kD}(\lambda) \rightarrow [0,1]\) is the compactly supported continous function given by
\[
\phi_{kD}(\lambda) = \mathrm{Vol}_{d-1}([W_{v} \times \{z=\lambda\}] \setminus \bigcup_{u \in \mathbb{Z}^{d-1}} [(u,1)+C_{kD}]),
\]
where \(e_{0}(\cdot)\) denotes the Hilbert-Samuel multiplicity, and for any vertex \(v \in \mathbb{Z}^{d-1}\) we denote by \(W_{v} \subset \mathbb{R}^{d-1}\) the \(d-1\) dimensional unit cell at the vertex \(v\).
Combining the above two results we get the main theorem of the paper under review.
Main Result. Let \(X\) be a projective toric variety with a very ample torus invariant Cartier divisor \(D\) with the associated homogeneous ring \((R,m)\). We have
\[
\lim_{k \rightarrow \infty} \frac{e_{H\, K}(R,m^{k}) - e_{0}(R,m^{k})/d!}{k^{d-1}} = \frac{e_{0}(R,m)}{(d-1)!} \int_{0}^{\infty} \phi_D{}(\lambda) d\lambda,
\]
where \(\phi_{D} : [0,\infty) \rightarrow [0,\infty)\) is defined as above (with \(k=1\)) and is solely determined by the shape of the polytope \(P_{D}\) associated with the toric pair \((X,D)\). tiling; Hilbert-Kunz multiplicity; convex geometry; Hilbert-Kunz density; projective toric varieties Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Tilings in \(n\) dimensions (aspects of discrete geometry) Hilbert-Kunz density function and asymptotic Hilbert-Kunz multiplicity for projective toric varieties | 0 |
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 crepant morphisms of algebraic threefolds. The main result of the paper (corollary 2, \S2) states that a birational map \(f:X\to Y\), of normal threefolds \(X\) and \(Y\) with only terminal singularities is always an isomorphism in codimension one, provided the canonical class \(K_ Y\) is nef. This result is obtained as a consequence of some results of \textit{Miles Reid} (resp. the \textit{weak index theorem}) and of some transitive properties of crepancy, proved by the author (see \S2). nef canonical class; crepant morphisms of algebraic threefolds; terminal singularities; weak index theorem Rational and birational maps, Minimal model program (Mori theory, extremal rays), \(3\)-folds A result on the birational map between two normal 3-folds | 0 |
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 studies a resolution of singularities for Drinfeld's relative compactification for moduli stacks of vector bundles over a connected smooth projective curve. When the curve has genus zero, a resolution of singularities was constructed in [\textit{B. Feigin} et al., ``Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces'', \url{arXiv:alg-geom/9711009}] for the space of quasimaps. Here the author similarly takes the approach of Kontsevich's stable maps from nodal deformations of the curve into twisted flag varieties to obtain the desired resolution of singularities. As an application, the author shows that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic at points sufficiently antidominant relative to their defect. resolution of singularities; Drinfeld's compactification; stable maps Geometric Langlands program: representation-theoretic aspects, Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic) A resolution of singularities for Drinfeld's compactification by stable maps | 0 |
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 a previous paper [Compos. Math. 63, 123-142 (1987; Zbl 0655.14007)] the author introduced the notion of the scheme of birational automorphisms Bir(X) of an algebraic variety X. In good cases (B-good) \(Bir(X)_{red}\) is a group scheme which acts birationally on some model of X, and is universal among such group schemes. It was shown that minimal models are B-good.
In the present paper the author verifies that any non-ruled variety is B- good. This agrees with a well-known conjecture that every non-ruled variety admits a minimal model. Other fundamental properties of Bir(X) verified in the case of minimal models are true also for non-ruled varieties. For instance, the author proves that dim(Bir(X))\(\leq \dim (X)\). scheme of birational automorphisms; minimal models; B-good Hanamura M.: Structure of birational automorphism groups, I: non-uniruled varieties. Invent. Math. 93, 383--403 (1988) Rational and birational maps, Minimal model program (Mori theory, extremal rays), Group actions on varieties or schemes (quotients) Stucture of birational automorphism groups. I: Non-uniruled varieties | 0 |
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 minimal free resolution of non-linearly normal embedded curves of high degree and the homogeneous ideal of general enough curves (for general \(k\)-gonal curves and, in a much wider range, for curves with general moduli). Then we consider the surjectivity of the Gaussian maps (or ``Wahl maps'') for non complete linear systems. Wahl maps; minimal free resolution of nonlinearly normal embedded curves; Gaussian maps; non complete linear systems --------, On the minimal free resolution of some projective curves , Ann. Mat. Pura Appl., Plane and space curves, Divisors, linear systems, invertible sheaves On the minimal free resolution of some projective curves | 0 |
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 \(\mathbb C^{n}\) we consider an algebraic surface \(Y\) and a finite collection of hypersurfaces \(\{S_{i}\}\). Froissart's theorem states that if \(Y\) and \(\{S_{i}\}\) are in general position in the projective compactification of \(\mathbb C^{n}\) together with the hyperplane at infinity then for the homologies of \(Y \setminus \bigcup S_{i}\) we have a special decomposition in terms of the homology of \(Y\) and all possible intersections of \(S_{i}\) in \(Y\). We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of \(\mathbb C^n\) in which \(Y\) and \(\{S_{i}\}\) are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in \(Y\) leaving invariant all hypersurfaces \(Y \cap S_{k}\) with the exception of one \(Y \cap S_{i}\), which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface \(Y\) the complement of a surface in a compact toric variety to a collection of hypersurfaces in it. homology group; toric variety; coboundary operator Toric varieties, Newton polyhedra, Okounkov bodies, Homotopy theory and fundamental groups in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry On isotopies and homologies of subvarieties of toric varieties | 0 |
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 \((V,0)\) be the germ of a normal surface singularity and \(\pi:X\to V\) its minimal resolution. Let \(L\) be a line bundle on \(X\) and \(H^0(X,L)\neq 0\). The fixed part of the linear system \(|L|\) is the biggest effective divisor \(F\) supported in \(\pi^{-1}(0)\) such that the restriction map \(H^0(X,L)\to H^0(F,L)\) is zero.
Assume \(L-K_X\) is nef. It is proved that the fixed part of \(|L|\) supports at most exceptional sets of rational singular points. If \((U,p)\) denotes the rational singular point obtained by contracting a connected component of the fixed part of \(|L|\) then the multiplicity \(\text{mult}(U,p)\) and the embedding dimension satisfy \(\text{mult}(U,p)\leq 2_{p_f}(V,0)\) and \(\text{embdim}(U,p)\leq 2_{p_f}(V,0)+1\). Here \(p_f(V,0)\) is the arithmetic genus of the fundamental cycle on \(\pi^{-1}(0)\). surface singularity; fixed component; reducible curve K. Konno, On the fixed loci of the canonical systems over normal surface singularities, Asian J. Math. 12 (2008), 449--464. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry On the fixed loci of the canonical systems over normal surface singularities | 0 |
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 the toric variety associated to the Bruhat poset of Schubert varieties in the Grassmannian. In this paper the authors describe the singular locus of \(X\) in terms of faces of the cone associated to \(X\). It is proved that the singular locus is pure of codimension \(3\) in \(X\), and the generic singularities are of cone type. They also determine the tangent cones at the maximal singularities of \(X\). In the case of \(X\) being associated to the Bruhat poset of Schubert varieties in the Grassmannian of \(2\)-planes in a \(n\)-dimensional vector space (over the base field), they show a certain product formula relating the multiplicities at certain singular points. Brown, J.\!; Lakshmibai, V.\!, Singular loci of Grassmann--Hibi toric varieties, Michigan Math. J., 59, 243-267, (2010) Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds Singular loci of Grassmann-Hibi toric varieties | 0 |
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 \textit{P. Ellia} and \textit{A. Hirschowitz} [J. Algebr. Geom. 1, No. 4, 531-547 (1992; Zbl 0812.14036)] showed that under certain conditions there exists a smooth, skew and connected curve \(C\) in 3-space whose ideal sheaf admits a linear resolution of a special type. The general curve with this property is unique and the corresponding irreducible component of the Hilbert scheme is unirational. In the present paper the author obtains similar results for curves whose sheaf of ideals admits a quasi-linear resolution as it is called by the author. space curve; ideal sheaf; Hilbert scheme; quasi-linear resolution Plane and space curves Curves in \(\mathbb{P}^3\) with quasi-linear resolution | 0 |
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 investigates a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. This method is based on the smoothing theorem of Kawamata and Namikawa. In particular, this method enables one to construct different Calabi--Yau threefolds with the same Hodge numbers. The author, using Clemens--Schmid sequence, describes, under some conditions (Condition 6.4), the Picard group of the smoothing in terms of Picard groups of components of the central fiber. Some applications are included, such as a construction of new examples of Calabi--Yau 3-folds with Picard number one with some interesting properties. Semistable degeneration; Calabi--Yau variety Lee N.-H.: Calabi--Yau construction by smoothing normal crossing varieties. Int. J. Math. 21(6), 701--725 (2010) Calabi-Yau manifolds (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry Calabi-Yau construction by smoothing normal crossing varieties | 0 |
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 discuss the resolution of toroidal orbifolds. For the resulting smooth Calabi-Yau manifolds, we calculate the intersection ring and determine the divisor topologies. In a next step, the orientifold quotients are constructed.. Lüst, D.; Reffert, S.; Scheidegger, E.; Stieberger, S., Resolved toroidal orbifolds and their orientifolds, Adv. Theor. Math. Phys., 12, 67, (2008) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects) Resolved toroidal orbifolds and their orientifolds | 0 |
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 rank 2 vector bundle, \(\mathcal E\), on \(\mathbb P^2\) has a free resolution of the form
\[
0 \rightarrow \bigoplus_{i=1}^k {\mathcal O}_{\mathbb P^2}(-a_i)@>\phi>>\bigoplus_{j=1}^{k+2} {\mathcal O}_{\mathbb P^2} (-b_j) \rightarrow {\mathcal E} \rightarrow 0.
\]
The authors study general elements of the moduli spaces \({\mathfrak M}_{\mathbb P^2} (2,c_1,c_2)\) of rank 2 bundles with Chern classes \(c_1, c_2\), together with their minimal free resolutions. \textit{G. Bohnhorst} and \textit{H. Spindler} [in: Complex algebraic varieties, Proc. Conf., Bayreuth/Ger. 1990, Lect. Notes Math. 1507, 39--50 (1992; Zbl 0773.14008)] have shown that the Betti numbers define a stratification of this moduli space by constructible subsets. Here the authors estimate the codimension of such strata, applying the results of Bohnhorst and Spindler, and characterize the Betti numbers of the general element of the moduli space. More specifically, they consider normalized semistable or stable rank 2 bundles on \(\mathbb P^2\) and their regularity. As a consequence, they get a new proof of the irreducibility of this moduli space. Dionisi, C.; Maggesi, M.: Minimal resolution of general stable vector bundles on P2, Boll. unione mat. Ital. 6, No. 1, 151-160 (2003) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli Minimal resolution of general stable rank-2 vector bundles on \(\mathbb P^2\) | 0 |
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 his Ph. D. thesis ``A Formula for the Gromov-Witten Invariants of Toric Varieties'' (Université Louis Pasteur, Strasbourg, 1999; Zbl 0964.14045), the author of the present paper has recently proven an explicit combinatorial formula for the genus-zero Gromov-Witten invariants of a smooth projective toric variety \(X\) with respect to the distinguished cohomology class \(\beta = 1\) in \(H^0(\overline{\mathcal M}_{0,m},\mathbb{Q})\), where \(\overline{\mathcal M}_{0,m}\) denotes the Deligne-Mumford compactification of the moduli space of genus-zero curves with \(m\) marked points.
In the paper under review, the author continues this study by deriving a similar formula for the case where the class \(\beta\) is the maximal product of Chern classes of cotangent lines to the marked points, that is for those classes that are Poincaré dual to a finite number of points in the moduli space \(\overline{M}_{0,m}\). At the end of the paper, this new formula is illustrated by explicitely exhibiting it for the special toric variety \(\mathbb{P}_{\mathbb{P}^1}({\mathcal O} \oplus(r-2)\oplus {\mathcal O}(1)\oplus {\mathcal O}(1))\). This is then used to compute the quantum cohomology ring of that special manifold, which provides an alternative approach to the one carried out earlier by \textit{V. V. Batyrev} in [Astérisque 218, 9-34 (1993; Zbl 0806.14041)]. Gromov-Witten invariants; quantum cohomology; toric varieties; symplectic manifolds; moduli spaces of stable curves; cohomology classes Spielberg, H.: Multiple quantum products in toric varieties, Int. J. Math. math. Sci. 31, No. 11, 675-686 (2002) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Multiple quantum products in toric varieties | 0 |
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 Given an algebraic variety \(V\) over a field \(k\) of characteristic zero, the following are two equivalent formulations of the Zariski-Lipman Conjecture.
\textsl{(1) Let \(p\) be a (closed) point of \(V\) with local ring \(R\). If the module of \(k\)-derivations \(\text{Der}_k(R)\) is a free \(R\)-module, then \(V\) is smooth at \(p\).}
(2) \textsl{If \(V\) is normal and the tangent bundle of the smooth locus of \(V\) is a trivial bundle, then \(V\) is smooth.}
The authors prove instances of (the second formulation of) the conjecture in dimension two, using the theory of non-complete algebraic surfaces by the Japanese school. More precisely, denote by \(V^0\) the smooth locus of an algebraic surface \(V\) and by \(\bar \kappa (V^0)\) its logarithmic Kodaira dimension. Assuming that moreover \(k\) is algebraically closed and that the tangent bundle of \(V^0\) is trivial, the authors show the following. {\parindent=6mm \begin{itemize} \item[(1)] If \(V\) is affine and \(\bar \kappa (V^0) \leq 1\), then \(V\) is smooth. \item [(2)] If \(V\) is projective, then \(\bar \kappa (V^0) \leq 0\) and \(V\) has at most one singularity. \item [(3)] If \(V\) is projective and \(\bar \kappa (V^0) =0\), then \(V\) is smooth.
\end{itemize}} Moreover, if \(V\) is projective and \(\bar \kappa (V^0) =-\infty\), they construct a \(\mathbb P^1\)- fibration \(W \to C\), where \(W\) is a resolution of singularities of \(V\) and \(C\) is a smooth projective curve. In this setting they `almost always\'\ prove that \(V\) is smooth, that is, except when this fibration has a certain very special form. Zariski-Lipman Conjecture; module of derivations; normal algebraic surfaces; non-complete algebraic surfaces Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Regular local rings, Derivations and commutative rings On the Zariski-Lipman conjecture for normal algebraic surfaces | 0 |
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 More than 50 years ago \textit{D. Mumford} gave an example of a generically non-reduced irreducible component of the Hilbert scheme of smooth irreducible space curves [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)]. Expanding on Mumford's example, \textit{J. O. Kleppe} used Hilbert flag schemes to show that if a general curve \(C\) in a Hilbert scheme component \(V\) lies on a smooth cubic surface \(X\) and \(H^1({\mathcal O}_X (-C)(3)) \neq 0\), then \(V\) is generically non-reduced [Lect. Notes Math. 1266, 181--207 (1985; Zbl 0631.14022)]. Recently \textit{J. O. Kleppe} and \textit{J. C. Ottem} extended these ideas to produce examples in which the general curve \(C\) lies on a quartic surface [Int. J. Math. 26, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)], but their method fails for curves lying on smooth surfaces \(X\) of degree \(d \geq 5\) or having Picard number \(\rho (X) > 2\).
Here the author combines Hilbert flag scheme methods and the theory of Hodge loci to construct more examples of non-reduced Hilbert schemes whose general curve \(C\) lies on a smooth surface \(X\) of degree \(d \geq 5\) (and on no quartic) satisfying \(\rho (X) > 2\). He starts with a smooth surface \(X\) of degree \(d \geq 5\) containing two coplanar lines \(L_1\) and \(L_2\), noting that \(\rho (X) > 2\). The Hilbert scheme of extremal curves of the form \(D=2L_1 + L_2 \subset X\) is generically non-reduced by work of \textit{M. Martin-Deschamps} and \textit{D. Perrin} [Ann. Scient. École Norm. Sup. 29, 757--785 (1996; Zbl 0892.14005)]. Letting \(\gamma \in H^{1,1}(X, \mathbb Z)\) denote the cohomology class of of \(D\), the Zariski closure of the associated Hodge locus \(\text{NL}(\gamma)\) in the open set \(U \subset |{\mathcal O}_{\mathbb P^3} (d)|\) of smooth surfaces is irreducible and generically non-reduced. With arguments involving exact sequences and Mumford regularity, he shows that the general member \(C \in |{\mathcal O}_X (D) (m)|\) is a smooth connected curve for \(m \geq 2d-2\). Letting \(X\) and \(C\) vary and taking the closure in the Hilbert scheme gives an irreducible component which is generically non-reduced. The arguments are clearly written and easy to follow. Hilbert scheme of space curves; Hilbert flag scheme; Hodge locus; non-reduced components Parametrization (Chow and Hilbert schemes), Transcendental methods of algebraic geometry (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) On generically non-reduced components of Hilbert schemes of smooth curves | 0 |
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 main object of the paper under review is a smooth compactification \(X\) of an anisotropic torus \(T\) defined over a number field \(K\). The authors are interested in asymptotical distribution of rational points \(X(K)\) with respect to the anticanonical height. Their main result gives such an asymptotic expressed in geometric terms which is in full accordance with a conjecture by \textit{V.~V. Batyrev and Yu.~I. Manin} [Math. Ann. 286, No. 1--3, 27--43 (1990; Zbl 0679.14008)] refined by \textit{E. Peyre} [Duke Math. J. 79, No. 1, 101--218 (1995; Zbl 0901.14025)].
Note that recently the authors obtained the same asymptotics for a smooth compactification of an arbitrary torus \(T\) [J. Algebr. Geom. 7, 15--53 (1998; Zbl 0946.14009)]. toric variety; anticanonical height; asymptotical distribution of rational points V. V. Batyrev and Y. Tschinkel, ''Rational points of bounded height on compactifications of anisotropic tori,'' Internat. Math. Res. Notices, iss. 12, pp. 591-635, 1995. Rational points, Toric varieties, Newton polyhedra, Okounkov bodies, Arithmetic varieties and schemes; Arakelov theory; heights Rational points of bounded height on compactifications of anisotropic tori | 0 |
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 article under review, the authors study the syzygies of curves lying in toric surfaces. In doing so, they deduce from their main result that:
\begin{itemize} \item[(i)] the canonical graded Betti table of a degree \(d \geq 5 + \lfloor \delta / 3 \rfloor\) curve in \(\mathbb{P}^2\) having \(\delta \leq 3\) nodes in general position depends on \(d\) and \(\delta\) only; \item[(ii)] the canonical graded Betti table of a smooth curve of weighted degree \(2d\) in \(\mathbb{P}(1,1,2)\), for some integer \(d \geq 3\), only depends on \(d\); and \item[(iii)] the canonical graded Betti table of smooth curves of weighted degree \(6d\) in \(\mathbb{P}(1,2,3)\), for some integer \(d \geq 2\), only depends on \(d\). \end{itemize}
As one additional result, the authors verify that the conclusion of M. Green's syzygy conjecture, for canonically embedded curves, holds true for all smooth non-hyperelliptic curves with genus at least \(32\) or Clifford index at most equal to six.
One aspect of the authors' approach is that if \(\Delta^{(1)}\) is the convex hull of the lattice points inside of the interior of a lattice polygon \(\Delta \subseteq \mathbb{R}^2\) and \(|\Delta^{(1)} \cap \mathbb{Z}| \geq 4\), then combinatorial properties of \(\Delta^{(1)}\) can be used to determine the Clifford index of curves lying in \(X_{\Delta}\), the toric surface which is determined by \(\Delta\).
This interplay between combinatorial aspects of \(\Delta^{(1)}\) and curves lying in \(X_{\Delta}\) has origins in work of \textit{A. G. Khovanskii} [Funct. Anal. Appl. 11, 289--296 (1978; Zbl 0445.14019); translation from Funkts. Anal. Prilozh. 11, No. 4, 56--64 (1977)]. algebraic curves; toric surfaces; syzygies Special algebraic curves and curves of low genus, Syzygies, resolutions, complexes and commutative rings, Toric varieties, Newton polyhedra, Okounkov bodies Canonical syzygies of smooth curves on toric surfaces | 0 |
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 variety \(X \subset \mathbb P^n\) over an algebraically closed field is said to have minimal degree if \(\deg X = 1 + n - \dim X\), and such varieties are completely classified. Among these are the rational normal scrolls (RNS). A RNS of dimension \(d\) is associated to a sequence of positive integers \(a_1,\dots,a_d\) and is denoted \(S(a_1,\dots,a_d)\). A general hyperplane section \(Y\) of \(X = S(a_1,\dots,a_d)\) is again a rational normal scroll, this time of dimension \(d-1\), so it is of the form \(S(b_1,\dots,b_{d-1})\). This paper gives a complete description of how the sequences \(a_1,\dots,a_d\) and \(b_1,\dots,b_{d-1}\) are related, and in so doing gives a complete classification of the RNS's that are hyperplane sections of a given RNS. The methods are both algebraic and geometric. rational normal scrolls; determinantal ideals Determinantal varieties, Linkage, complete intersections and determinantal ideals A remark on hyperplane sections of rational normal scrolls | 0 |
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 main object of the paper under review is a \(\Delta\)-regular [in the sense of \textit{V.~V. Batyrev}, J. Algebr. Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)] divisor \(X\) on a proper toric variety \(V\) (defined by a fan \(\Delta\)). The ground field \(k\) is assumed to be algebraically closed. The author's goal is to prove for \(X\) the minimal model conjecture: If the Kodaira dimension \(\kappa (X)\) is non-negative then \(X\) has a minimal model \(Y\) with abundance (i.e. the linear system \(|mK_Y|\) is base point free for \(m\) big enough), and if \(\kappa (X)=-\infty\) then \(X\) is birationally equivalent to a proper variety \(Y\) with at worst terminal singularities, with the relatively ample anti-canonical divisor, and admitting a fibration to a lower dimensional variety. This conjecture is proved using the techniques of \textit{M.~Reid} [in: Arithmetic and geometry, Vol. II: Geometry, Prog. Math. 36, 395-418 (1983; Zbl 0571.14020)] (successive contractions of extremal rays and flips). If \(\operatorname {char} k=0\), by Bertini's theorem this implies the minimal model conjecture for a general member of a linear base point free system on any proper toric variety. Moreover, for \(X\) with \(\kappa (X)\geq 0\) the author presents a concrete algorithm for constructing a minimal model with abundance by means of the polytope of the adjoint divisor. toric variety; \(\Delta\)-regular divisor; minimal model conjecture; extremal rays; flips Ishii S. The minimal model theorem for divisors of toric varieties. Tohoku Math J (2), 1999, 51: 213--226 Toric varieties, Newton polyhedra, Okounkov bodies, Minimal model program (Mori theory, extremal rays) The minimal model theorem for divisors of toric varieties | 0 |
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 \(\pi : (\tilde X, E) \longrightarrow (X, o)\) be a normal surface singularity \((X, o)\). For an element \(h \in \mathfrak m_{X,o} \subset \mathcal O_{X,o}\), we consider a cycle \(\sum\limits^{r}_{j=1} v_{E_j} (h \circ \pi) E_j\) on \(E\), where \(E = \bigcup\limits^{r}_{j=1} E_j\) is the irreducible decomposition of \(E\). Among all such cycles, there is the minimal one \(M_E\) (S. S. T.Yau). It is called the maximal ideal cycle on \(E\). In generally, we have \(Z_{E} \leqq M_{E}\) for the fundamental cycle \(Z_E\). In the normal surface singularity theory, it is a natural problem to compare \(M_E\) and \(Z_E\). There are several results on it (see references of this paper).
If \((X, o)\) is a hypersurface singularity defined by \(z^{a_1}_{1} +\cdots + z^{a_n}_{n} = 0 (a_j \geqq 2\) for any \(j\)), then it is called a hypersurface singularity of Brieskorn type. In this paper, for 2-dimensional hypersurface singularities of Brieskorn type (i.e., \(n = 3\)), the authors computed the maximal ideal cycle for the minimal good resolution of \((X, o)\) by considering a cyclic covering of \(\mathbb C^2\). Namely, they determined the \(M_E\) from \(a, b, c\), and compared \(M_E\) and \(Z_E\) . They obtain a necessary and sufficient condition to coincide them. Konno, K., Nagashima, D.: Maximal ideal cycles over normal surface singularities of Brieskorn type. Osaka J. Math. 49, 225-245 (2012) Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Maximal ideal cycles over normal surface singularities of Brieskorn type | 0 |
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 \(V/\Gamma\) be a symplectic quotient singularity, i.e. a quotient of a finite dimensional vector space \(V\) by a linear action of a finite subgroup \(\Gamma \subset \mathrm {Sp}(V)\). The article under review concerns \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), i.e. projective, crepant, birational morphisms \(\rho : Y \rightarrow V/\Gamma\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities, and symplectic resolutions of \(V/\Gamma\), i.e. smooth \(\mathbb{Q}\)-factorial terminalizations. By results of Namikawa, a symplectic quotient singularity admits finitely many \(\mathbb{Q}\)-factorial terminalizations, and if one of them is smooth then all are smooth.
The main result is a formula for the number of non-isomorphic \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), expressed in terms of the Calogero-Moser deformation of \(V/\Gamma\) and the Namikawa Weyl group associated to \(V/\Gamma\). From this theorem the author derives a more explicit formula or obtains a number of symplectic resolutions for all groups \(\Gamma\) such that \(V/\Gamma\) is known to admit a symplectic resolution. These are: the infinite series of wreath products \(\mathcal{S}_n \wr G\), where \(G \subset \mathrm {SL}(2,\mathbb{C})\), acting on \(V = \mathbb{C}^{2n}\), a 4-dimensional representation \(G_4\) of the binary tetrahedral group and a 4-dimensional representation of \(Q_8 \times_{\mathbb{Z}_2} D_8\). In the case of \(\mathcal{S}_n \wr G\) the result is a formula involving the description of the Weyl group associated to \(G\) via the McKay correspondence, and for the remaining two cases the author obtains 2 and 81 symplectic resolutions respectively. symplectic resolutions; symplectic reflection algebras; Orlik-Solomon algebras Bellamy, G., Counting resolutions of symplectic quotient singularities, Compos. Math., 152, 1, 99-114, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Deformations of associative rings, Poisson algebras Counting resolutions of symplectic quotient singularities | 0 |
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 concerns the dimensions of certain affine Deligne-Lusztig varieties. We first set up the necessary notations.
Let \(L = k((t))\) and \(F = \bar{k}((t))\) for a finite field \(k\). Let \(\sigma\) denote the Frobenius automorphism of \(\bar{k}/k\) and extend it in a natural way to an automorphism of \(F/L\) by demanding \(\sigma(t) = t\). We denote the valuation ring of \(L\) by \({\mathcal O}_L\). Let \(G\) be a split connected reductive \(k\)-group, \(A\) a split maximal torus of \(G\) and \(K = G({\mathcal O}_L) \subseteq G(L)\). For \(b \in G(L)\) and for a dominant \(\mu \in X_*(A)\), the \textit{affine Deligne-Lusztig variety} is the locally closed subscheme of the affine Grassmannian, \(X := G(L)/K\), defined by
\[
X_{\mu}(b) := \{x \in G(L)/K: x^{-1}b\sigma(x) \in K\mu(t)K\} .
\]
\textit{M. Rapoport} [Astérisque 298, 271--318 (2005; Zbl 1084.11029)] conjectured a formula for the dimensions of these affine Deligne-Lusztig varieties which was modified later by \textit{R. Kottwitz} [Pure Appl. Math. Q. 2, No.~3, 817--836 (2006; Zbl 1109.11033)]. Let \(\mathbb D\) be a torus over \(F\) with character group \(\mathbb Q\). The element \(b\) determines a map \(\nu_b : {\mathbb D}\to G\) and the associated coweight \(\bar{\nu}_b \in X_*(A)_{\mathbb Q}\) is dominant. Let \(M_b\) denote the centraliser of \(\nu_b({\mathbb D})\) in \(G\). There exists an inner form \(J\) of \(M_b\) such that \(J(R) = \{g \in G(R \otimes_F L): g^{-1}b\sigma(g) = b\}\) for any \(F\)-algebra \(R\). If \(X_{\mu}(b)\) is nonempty, then the conjectured dimension formula is
\[
\dim X_{\mu}(b) = \langle \rho, \mu-\bar{\nu}_b \rangle-\frac{1}{2}(\text{rk}_F(G)-\text{rk}_F(J))
\]
where \(\rho \in X^*(A)_{\mathbb Q}\) denotes the half sum of the positive roots. It is proved in this paper that if this dimension formula holds for ``superbasic'' elements \(b \in\text{GL}_n(L)\), then it holds in general. This particular case is recently proved by \textit{E. Viehmann} [Ann. Sci. Éc. Norm. Supér. (4) 39, No.~3, 513--526 (2006; Zbl 1108.14036)] so now the conjecture is completely solved.
The paper also investigates the dimensions of the affine Deligne-Lusztig varieties in the affine flag manifold \(G(L)/I\) where \(I\) is the Iwahori subgroup of \(G(L)\) obtained from an alcove \({\mathbf a}_1\) in the apartment associated to \(A\). Deligne-Lusztig varieties; reductive groups Görtz, U.; Haines, T. J.; Kottwitz, R. E.; Reuman, D. C., Dimension of certain affine Deligne-Lusztig varieties, Ann. Sci. Ecole Norm. Sup. (4), 39, 467-511, (2006) Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over local fields and their integers, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties Dimensions of some affine Deligne-Lusztig varieties | 0 |
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 There is a well-developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite, the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space, extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks. D. Edidin, Y. More, in: \textit{Integration on Artin Toric Stacks and Euler Characteristics}, Proc. Amer. Math. Soc., to appear, arXiv:1201.2104. Toric varieties, Newton polyhedra, Okounkov bodies, (Equivariant) Chow groups and rings; motives, Stacks and moduli problems Integration on Artin toric stacks and Euler characteristics | 0 |
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 \(X_{\operatorname{\Sigma}}\) be a smooth complete toric variety defined by a fan \(\Sigma\) and let \(V = V(I)\) be a subscheme of \(X_{\operatorname{\Sigma}}\) defined by an ideal \(I\) homogeneous with respect to the grading on the total coordinate ring of \(X_{\operatorname{\Sigma}}\). We show a new expression for the Segre class \(s(V, X_{\operatorname{\Sigma}})\) in terms of the projective degrees of a rational map specified by the generators of \(I\) when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where \(X_{\operatorname{\Sigma}}\) is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of \(X_{\operatorname{\Sigma}}\) via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class \(s(V, X_{\operatorname{\Sigma}})\), the Chern-Schwartz-MacPherson class \(c_{S M}(V)\) and the topological Euler characteristic \(\chi(V)\) of \(V\). These algorithms can, in particular, be used for subschemes of any product of projective spaces \(\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_j}\) or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms. Euler characteristic; Chern-Schwartz-MacPherson class; Segre class; Chern class; toric varieties; computer algebra; computational intersection theory Helmer, M.: Computing characteristic classes of subschemes of smooth toric varieties. J. algebra (2017) Toric varieties, Newton polyhedra, Okounkov bodies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation Computing characteristic classes of subschemes of smooth toric varieties | 0 |
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 the toroidal resolution of a plane curve \(C=\{f(z,w) = 0\}\), which is a resolution consisting of a finite composition of admissible toric blowing-ups. It is shown how to read Puiseux pairs and the intersection multiplicities among irreducible components using the data of the toroidal resolution. The results are applied to planar curves, which are obtained as an \(n\)-times iterated generic hyperplane section of a non-degenerated hypersurface, to get that:
(1) the resolution complexity of these curves are at most \((n+1)\);
(2) each irreducible component of the curve has at most one Puiseux pair; and
(3) no Puiseux pair implies smoothness. toroidal resolution of a plane curve; toric blowing ups; Puiseux pair M. Oka: Geometry of plane curves via toroidal resolution ; in Algebraic Geometry and Singularities (La Rábida, 1991), Progr. Math. 134 , Birkhäuser, Basel, 95-121, 1996. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Geometry of plane curves via toroidal resolution | 0 |
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 concerned with the following conjecture:
Conjecture. Let \(f:\mathbb{C}^n\rightarrow \mathbb{C}^{n+1}\) be an injective polynomial map such that \(V:=f(\mathbb{C}^n)\) is a smooth algebraic subvariety of \(\mathbb{C}^{n+1}\). Then there exists an automorphism \(\varphi\) of \(\mathbb{C}^n\) that rectifies \(V\), i.e. \(\varphi(V)=\mathbb{C}^n\times \{0\}\).
The case \(n=1\) is the celebrated Abhyankar-Moh-Suzuki theorem. The paper under review proves a special case of \(n=2\). For this, the author defines \(X\) as the closure of \(f(\mathbb{C}^2)\) in \(\mathbb{P}^3\) (under the canonical embedding of \(\mathbb{C}^3\) in \(\mathbb{P}^3\)), and \(Y:=X\backslash f(\mathbb{C}^2)\). In the case that \(X\) is a normal quartic hypersurface without triple points, he proves the conjecture and describe how to find a rectifying automorphism.
[For part I, cf. Osaka J. Math. 38, No. 3, 507--532 (2001; Zbl 1010.32012)]. algebraic embedding Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Compactification of analytic spaces The structure of algebraic embeddings of \(\mathbb{C}^{2}\) into \(\mathbb{C}^{3}\) (the normal quartic hypersurface case. II) | 0 |
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 shows that certain nilpotent orbits have normal closures. For the classical groups, Kraft and Procesi solved almost all cases of the question of which nilpotent orbits have normal closures and which do not. The remaining cases are when the group is of type \(D_{2l}\) and the Jordan blocks of the elements of the orbit are of the form \((a^{2k},b^2)\), where \(a\) and \(b\) are even, \(a>b\) and \(ak+b=2l\). In the present article it is shown that the closures of these orbits are normal, and thus normality for all cases of nilpotent orbit closures for the classical groups is known. The method of proof is as follows. To check that the closure of an orbit \({\mathcal O}\) is normal, one finds another nilpotent orbit \({\mathcal O}'\) that is known to have a normal closure, and then to show that the ring of regular functions on \({\mathcal O}\) is a quotient of the ring of regular functions on \({\mathcal O}'\). This idea was previously used by the author in [\textit{E. Sommers}, J. Algebra 270, 288--306 (2003; Zbl 1041.22009)]. nilpotent orbits; normal orbit closures Sommers, E., Normality of very even nilpotent varieties in \(D_{2 l}\), Bull. lond. math. soc., 37, 3, 351-360, (2005) Classical groups (algebro-geometric aspects), Vanishing theorems in algebraic geometry, Linear algebraic groups over the reals, the complexes, the quaternions Normality of very even nilpotent varieties in \(D_{2l}\) | 0 |
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 studies Homological Mirror Symmetry for toric del Pezzo surfaces, one of the few settings where both sides of the Mirror and a functor between them can be studied explicitly.
There are two broad approaches to Homological Mirror Symmetry in this context. The first goes back to the construction, due to Seidel, of a Fukaya category associated to a Picard-Lefschetz fibration. Using this category it is possible to compare DG categories generated by collections of Lagrangian vanishing cycles with known exceptional collections of sheaves in the del Pezzo surfaces. This approach was pursued in detail (for all del Pezzo surfaces and weighted projective planes) by Auroux-Katzarkov-Orlov [\textit{D. Auroux} et al., Ann. Math. (2) 167, No. 3, 867--943 (2008; Zbl 1175.14030)]. Another approach in the toric setting was taken by Abouzaid, in which one considers Lagrangian sections and, in the spirit of the SYZ conjecture, compares these with the category of line bundles on the mirror side.
The main result of the paper provides a comparison between these two approaches, in particular, a map from the critical locus of the mirror potential for each toric del Pezzo surface to the pre-category of tropical sections considered by Abouzaid. One useful guiding principle in studying mirror symmetry for toric varieties is the identification of some combinatorial aspect of the fan with a geometric structure on each side of the mirror. The main result provides an instance of this principle, using on a comparison between torus fixed points of a toric variety and certain classes of loops in the space of Laurent polynomials with appropriate Newton polytope.
Sections 2 and 3 of the paper concisely recall the necessary material from the theory of exceptional collections in this context, as well as Abouzaid's Homological Mirror Symmetry functor. These sections are clear and a useful, though brief, guide to some of the main features of the respective subjects.
The remainder of the paper is devoted to the explicit construction of the map from the critical locus of the potential to a tropical Lagrangian section in each case. Each of these examples involves a certain amount of explicit calculation, although they are simple enough to convey clearly the construction of the corresponding Lagrangian brane. toric del Pezzo surfaces; homological mirror symmetry Mirror symmetry (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Rational and ruled surfaces, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces | 0 |
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 fan-like simplicial sphere of dimension \(n-1\) such that its associated complete fan is strongly polytopal, and let \(v\) be a vertex of \(K\). Let \(K(v)\) be the simplicial wedge complex obtained by applying the simplicial wedge operation to \(K\) at \(v\), and let \(v_0\) and \(v_1\) denote two newly created vertices of \(K(v)\). In this paper, we show that there are infinitely many strongly polytopal fans \(\Sigma\) over such \(K(v)\)'s, different from the canonical extensions, whose projected fans \(\operatorname{Proj}_{v_i} \Sigma(i=0,1)\) are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such \(K(v)\)'s such that toric varieties over the underlying projected complexes \(K_{\operatorname{Proj}_{v_i} \Sigma}(i=0,1)\) are also projective. simplicial complexes; strongly polytopal; simplicial wedge operation; projective toric varieties; linear transforms; Gale transforms; Shephard's diagrams; Shephard's criterion Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Gale and other diagrams Simplicial wedge complexes and projective toric varieties | 0 |
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 See the joint review of part I, II in [Zbl 0122.38603]. Global theory and resolution of singularities (algebro-geometric aspects) Resolution of singularities of an algebraic variety over a field of characteristic zero. II | 0 |
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 1989, \textit{G. Ellingsrud} and \textit{C. Peskine} proved that there are only finitely many integral families of smooth surfaces in \(\mathbb{P}^4\) which are not of general type [cf. Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009)]. A similar result for families of 3-folds in \(\mathbb{P}^5\) was shown by \textit{R. Braun}, \textit{G. Ottaviani}, \textit{M. Schneider} and \textit{F. O. Schreyer} [in: Complex Analysis and Geometry, Univ. Ser. Math. 311-338 (1992; Zbl 0798.14023)].
In the paper under review, the authors describe how one can adapt the proofs of those results to include the case of families of varieties with at least two-codimensional singular locus and ``bounded non-normality''. With this last terminology they mean finite sets of data bounding the difference between the Hilbert polynomials of the singular varieties and their normalizations. The paper relies heavily on its two predecessors. smooth surfaces which are not of general type; two-codimensional singular locus; Hilbert polynomials Families, moduli, classification: algebraic theory, Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry Exposé A: On projective properties of varieties with smooth normalization | 0 |
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 compact surface S with ordinary singularities in a compact complex analytic threefold W, Kodaira defined the sheaf \(\Phi\) of germs of infinitesimal equisingular displacements (hence \(H^ 0(S,\Phi_ S)\) is the space of (global) infinitesimal equisingular displacements of S in W) [\textit{K. Kodaira}, Ann. Math., II. Ser. 74, 591-627 (1961; Zbl 0192.184)]. Since the normalization X of S is nonsingular, we can define a canonical map \(c: H^ 0(S,\Phi_ S)\to H^ 1(X,T_ X)\) where the latter is considered to be the space of infinitesimal deformations of X.
The author gives two cases when this map c is shown to be surjective, and calculates the number of moduli for an example of Enriques surfaces realized as singular surfaces of degree 6 in \({\mathbb{P}}^ 3\) with tetrahedral double curves. One of the cases is as follows: \(H^ 1(W,T_ W)=H^ 2(W,T_ W(-\log S))=H^ 1(W,\Omega^ 1_ W([S+K_ W]- \Delta))=H^ 2(W,\Omega^ 1_ W([S+K_ W]-\Delta))=H^ 0(X,T_ X)=0\) where \(\Delta\) is the double curve on S, and for a locally free sheaf \({\mathcal E}\), \({\mathcal E}(-\Delta)\) denotes the subsheaf of sections vanishing at \(\Delta\). - For the proof the author uses the method to study deformations of holomorphic maps developed by \textit{E. Horikawa} [J. Math. Soc. Japan 25, 372-396 (1973; Zbl 0254.32022)].
The article contains not a few typographical mistakes and unusual English phrases, which might irritate some readers. normalizations of surfaces with ordinary singularities; characteristic map; infinitesimal deformations; number of moduli; Enriques surfaces; deformations of holomorphic maps S. TSUBOI, On the number of moduli of non-singular normalizations of surfaces wit ordinary singularities, ibid. 32 (1983), 23-46. Families, moduli, classification: algebraic theory, Global theory and resolution of singularities (algebro-geometric aspects), Complex-analytic moduli problems, Singularities of surfaces or higher-dimensional varieties, Deformations of submanifolds and subspaces, Algebraic moduli problems, moduli of vector bundles On the number of moduli of non-singular normalizations of surfaces with ordinary singularities | 0 |
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 new proof of Hironaka's theorem on resolution of singularities is given. There are already several different constructive approaches [cf. \textit{E. Bierstone} and \textit{P. Milman}, Invent. Math. 128, 207--302 (1997; Zbl 0896.14006) or \textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér., IV. Sér 22, 1--32 (1989; Zbl 0675.14003)]. The resolution process is based on the choice of an invariant which measures the singularities and drops under blowing up the maximal stratum of this invariant. The choice of the invariant and the way to compute it makes the difference between the approaches to resolve singularities. The definition of the invariant is quite involved. It is defined inductively using the knowledge of the resolution process up to this moment. The induction defining the invariant is given by the intersection of the variety with a so-called hypersurface of maximal contact. The choice of this hypersurface is not canonical and some effort is needed to define the invariant in a canonical way.
The approach of this paper is similar to the approach of Bierstone and Milman but based additionally on two observations. The resolution process defined as a sequence of blowing-ups the ambient spaces can be applied simultaneously to a class of equivalent singularities obtained by simple modifications, i.e. to resolve a singularity it is allowed to tune it before starting: In the equivalence class a convenient representative given by a so-called homogenized ideal is chosen. The restrictions of homogenized ideals to different hypersurfaces of maximal contact define locally analytically isomorphic singularities. resolution of singularities; algorithmic resolution Włodarczyk, Jarosław, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc., 18, 4, 779-822, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Simple Hironaka resolution in characteristic zero | 0 |
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 an integral and non-degenerate variety and \(V\subset \mathbb {P}^r\) a \(k\)-dimensional linear subspace. For any \(P\in \mathbb {P}^r\) the \(X\)-rank of \(P\) is the minimal cardinality of a subset of \(X\) whose linear span contains \(P\). Here we study several integers related to the \(X\)-rank of \(V\), e.g. the minimal \(X\)-rank of all elements of a basis of \(V\) or the minimal sum of the \(X\)-ranks of a basis of \(V\). \(X\)-rank; linear subspace; Grassmannian Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Ranks of linear subspaces with respect to an embedded variety | 0 |
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 normal affine variety over the complex numbers with an effective action by the \((n-1)\)-dimensional torus \(T \simeq (\mathbb{C}^\star)^{n-1}\). We associate to \(T\) the lattice \(M \simeq \mathbb{Z}^{n-1}\) of characters and the dual lattice \(N=\mathrm{Hom}(M,\mathbb{Z})\) of one-parameter subgroups. The aim of the article under review is to compare two sets of combinatorial data associated with \(X\).
\textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat} [Toroidal embeddings. I. Lecture Notes in Mathematics. 339. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0271.14017)] have shown that, for certain open subsets \(U\) of a complete nonsingular curve \(C\), there exists a toroidal embedding \((U \times T,X'')\), where \(X''\) is the normalization of the graph of a rational quotient map from \(X\) to \(C\). This determines a combinatorial datum, namely the toroidal fan \(\Delta(X,U)\). It is a collection of cones in different lattices \(\mathbb{Z} \times N\), one for each point \(P \in C \setminus U\), glued along their common face in \((0,N)\).
On the other hand, \textit{K. Altmann} and \textit{J. Hausen} [Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060)] gave a description of \(X\) by a divisor \(D\) with polyhedral coefficients on a nonsingular curve \(Y\). The divisor \(D\) is of the form \(\sum_{P \in Y}\Delta_{P} \otimes P\), where the \(\Delta_{P}\) are polyhedra in \(N \otimes Q\) with tail cone \(\sigma\).
To compare these data, it is noted that the curve \(Y\) is an open subset of \(C\). Defining the homogenization of a polyhedron \(\Delta \subset N \otimes Q\) with tail \(\sigma\) to be the cone in \(\mathbb{Z} \times N\) generated by \((1,\Delta)\) and \((0,\sigma)\), the author obtains the main result of this paper: The toroidal fan \(\Delta(X,U)\) is equal to the fan obtained by gluing the homogenizations of the coefficient polyhedra \(\Delta_P\) of points \(P \in Y \setminus U\) along their common face \((0,\sigma)\). R Vollmert, Toroidal embeddings and polyhedral divisors, Int. J. Algebra 4 (2010) 383 Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients) Toroidal embeddings and polyhedral divisors | 0 |
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 was (when it first appeared, as an electronic preprint, in 2007 [\url{arXiv:0709.1529}]) the first step towards the proof of the conjecture of \textit{M. Boij} and \textit{J. Söderberg} [J. Lond. Math. Soc., II Ser. 78, No. 1, 85--106 (2008; Zbl 1189.13008)] concerning graded Betti numbers of Cohen-Macaulay modules over polynomial rings, a proof accomplished by \textit{D. Eisenbud} and \textit{F.-O. Schreyer} [J. Am. Math. Soc. 22, No. 3, 859--888 (2009; Zbl 1213.13032)]. The first part of this conjecture asserts that if \(A = K[x_1, \dots ,x_n]\) is a polynomial ring over a field \(K\) and if \(d_0 < d_1 < \dots < d_n\) is an increasing sequence of integers then there exists an exact complex of graded free \(A\)-modules of the following form:
\[
0 \longrightarrow A(-d_n)^{b_n} \longrightarrow \cdots \longrightarrow A(-d_0)^{b_0}
\]
for some integers \(b_0 > 0, \dots ,b_n > 0\) (actually, for any such complex, \(d_0, \dots ,d_n\) and \(b_0\) uniquely determine \(b_1, \dots ,b_n\)).
In the paper under review, the authors construct, assuming \(\text{char}\, K = 0\), a \(\text{GL}(n)\)-equivariant complex with the above mentioned properties. In order to state their results, one has to recall some facts concerning the representation theory of the general linear group. A \textit{composition} is a sequence of non-negative integers \(\mu = (\mu_1, \dots ,\mu_n) \in {\mathbb N}^n\). If \(\mu_1 + \cdots + \mu_n = k\), one says that \(\mu\) is a composition of \(k\). A \textit{partition} is a composition \(\lambda = (\lambda_1 ,\dots ,\lambda_n)\) with \(\lambda_1 \geq \dots \geq \lambda_n\). Let \(E\) be an \(n\)-dimensional \(K\)-vector space. To each partition \(\lambda\) one can associate an irreducible representation \(S^\lambda E\) of the group \(\text{GL}(E) \simeq \text{GL}(n)\). \(S^\lambda E\) is a subrepresentation of \(S^{\lambda_1}E\otimes \cdots \otimes S^{\lambda_n}E\). Actually, there is a result asserting that if one denotes by \(c_{p,q}\) the composite map:
\[
S^pE \otimes S^qE \longrightarrow S^pE \otimes E \otimes S^{q-1}E \longrightarrow S^{p+1}E \otimes S^{q-1}E
\]
then \(S^\lambda E\) is the intersection of the kernels of the maps \(\text{id}_{\, S^{\lambda_1}} \otimes \cdots \otimes c_{\lambda_i,\lambda_{i+1}} \otimes \cdots \otimes \text{id}_{\, S^{\lambda_n}}\). \textit{Pieri's Formula} asserts that
\[
S^\lambda E \otimes S^eE \simeq \bigoplus_\mu S^{\lambda + \mu}E
\]
where the sum is taken over the compositions \(\mu\) of \(e\) such that \(\lambda_{i-1} \geq \lambda_i + \mu_i\), \(i = 2, \dots ,n\).
Now, assuming \(d_0 \geq 0\), put \(e_0 = d_0\), \(e_i = d_i - d_{i-1}\), \(i = 1,\dots ,n\), and \(\lambda_i = e_0 + \sum_{j = i+1}^n (e_j - 1)\), \(i = 0,\dots ,n\). Consider the partitions \(\alpha(0) = (\lambda_1, \dots , \lambda_n)\) and \(\alpha(i) = (\lambda_1 + e_1 , \dots ,\lambda_i + e_i, \lambda_{i+1}, \dots ,\lambda_n)\), \(i = 1 ,\dots ,n\). Using Pieri's Formula one verifies easily that there exist unique non-zero \(\text{GL}(E)\)-equivariant maps
\[
S^{\alpha(i)}E \longrightarrow S^{\alpha(i-1)}E \otimes S^{e_i}E,\;i\geq 1 \, ,
\]
and that the composite maps
\[
S^{\alpha(i+1)}E \rightarrow S^{\alpha(i)}E \otimes S^{e_{i+1}}E \rightarrow S^{\alpha(i-1)}E \otimes S^{e_i}E \otimes S^{e_{i+1}}E \rightarrow S^{\alpha(i-1)}E \otimes S^{e_i + e_{i+1}}E
\]
are zero. One constructs, in this way, a \(\text{GL}(E)\)-equivariant complex \(0 \rightarrow F_n \rightarrow \cdots \rightarrow F_0\) of graded free \(A\)-modules, with \(F_i = S^{\alpha(i)}E \otimes_KA(-d_i)\). Here, \(A\) is the symmetric algebra \(S(E) \simeq K[x_1, \dots, x_n]\). The main result of the paper under review asserts that this complex is exact. Despite the elementary character of the statement, the proof is quite involved. It uses geometric arguments based on a special case of Bott's theorem.
The authors also provide a second construction of a pure complex with given degree shifts \(d_0 < \dots < d_n\), resolving a Cohen-Macaulay module supported on the degeneracy locus of a generic map of free modules \(G \rightarrow F\) of ranks \(n\) and \(m\), respectively, over the polynomial ring \(B = k[(x_{ij} \, | \, 1\leq i \leq m,1 \leq j \leq n)]\). This complex is \(\text{GL}(m) \times \text{GL}(n)\)-equivariant. The authors show, moreover, that both results hold in the more general context of complexes of graded free modules over rings of the form \(A\otimes_K\Lambda\), where \(A\) is a polynomial ring and \(\Lambda\) is an exterior algebra. pure resolution; equivariant resolution; Betti diagram; Boij-Söderberg theory; Pieri map; determinantal variety Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy, The existence of equivariant pure free resolutions, Ann. Inst. Fourier, 61, 905-926, (2011) Syzygies, resolutions, complexes and commutative rings, Cohen-Macaulay modules, Determinantal varieties, Representation theory for linear algebraic groups The existence of equivariant pure free resolutions | 0 |
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 An \(n\)-dimensional normal variety \(X\) over \(\mathbb C\) on which \((\mathbb C^*)^n\) algebraically such that \(X\) contains a dense open orbit \(U\) which is equivariantly isomorphic to \((\mathbb{C}^*)^n\) where the group \((\mathbb C^*)^n\) acts on itself by translation is called a toric variety. A smooth complete toric variety is called a toric manifold. It is known that there are toric manifolds which are not projective manifolds when \(n\geq 3\), although when \(n=2\) any toric manifold is projective.
Generalizing the notion of a projective toric manifold, \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)] introduced the notion of what they called toric manifolds, which are now referred to as quasitoric manifolds. A real \(2n\)-dimensional smooth compact manifold \(M\) which is acted on by the \(n\)-dimensional (compact) torus \(T=(\mathbb S^1)^n\) is a quasitoric manifold if the action is `locally standard' and if the quotient space \(M/T\) is a simple polytope. Every projective toric manifold is a quasitoric manifold and it is known that there are quasitoric manifold which are not projective toric manifolds. The restricted action of \(T\subset (\mathbb{C}^*)^n\) on a toric manifold \(X\) satisfies the local standardness condition. The author shows, in Theorem I, that there are infinitely many toric manifolds \(X\) of dimension \(n\) (over \(\mathbb C\)) such that \(X/T\) is not a simple polytope, for every \(n\geq 4\). Such toric manifolds cannot be quasitoric. This answers a question raised by \textit{V. M. Buchstaber} and \textit{T. E. Panov} in their book [Torus actions and their applications in topology and combinatorics. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1012.52021)].
From the introduction: ``In a 2003 preprint [``Some examples in toric geometry'', \url{arXiv:math/0306029}] \textit{Y. Civan} claimed the existence of a toric manifold as in Theorem I, but it has been recognized that his explanation of non-polytopality was insufficient. Our construction is based on ideas found in (his preprint) but is more explicit.''
Recall that a simplicial sphere (i.e., a simplicial complex homeomorphic to a sphere) is called \textit{polytopal} if it is (simplicially isomorphic to) the boundary complex of a simplicial polytope. The dual (or polar) of a simplicial polytope is a simple polytope and conversely.
An example of a simplicial sphere which is not polytopal is the Barnette sphere \(\mathcal B\), which is \(3\)-dimensional---the smallest possible dimension in which a non-polytopal sphere can exist. It has been shown by Ishida, Fukukawa, and Masuda [\textit{H. Ishida} et al., Mosc. Math. J. 13, No. 1, 57--98 (2013; Zbl 1302.53091)] that the Barnette sphere cannot arise as the underlying simplicial sphere of a regular fan. The author first finds a simplicial fan in dimension \(4\) inducing the Barnette sphere \(\mathcal B\) and finds a suitable subdivision of the fan which is regular (and corresponds to a subdivision \(\mathcal B'\) of the Barnette sphere). The corresponding complete toric variety \(X\) is smooth but the quotient \(X/T=P\) cannot be a simple polytope. (If \(P\) were a simple polytope, its dual \(Q:=P^\circ\) would be simplicial and so \(\mathcal B'=\partial Q\) would then be polytopal.) To obtain infinitely infinitely many examples, one performs repeated blow ups. To obtain examples in higher dimensions, one takes repeated suspensions. fan; toric manifold; quasitoric manifold; Barnette sphere Mauranda, N; Barrere, V, Prototype for the optimization of CO2 injection wells placement in a reservoir, Energy Procedia, 63, 3097-3106, (2014) Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Compact Lie groups of differentiable transformations Examples of smooth compact toric varieties that are not quasitoric manifolds | 0 |
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 paper there is given a formula for the calculation of the Picard group of an arbitrary compact toric variety \(X_ \Sigma\). This improves the standard result in that \(X_ \Sigma\) may be singular and a previous result of the author in that \(X_ \Sigma\) may be non-projective. The main difference between the singular and the nonsingular case is that in the nonsingular case the rank of the Picard group of the variety \(X_ \Sigma\) is known to be determined by the combinatorial structure of \(\Sigma\) whereas in the singular case \(\text{Pic} X_ \Sigma\) may depend on metrical properties of the fan.
The proof of the formula is based on the geometric description of special \(T\)-invariant (i.e. invariant under the action of the torus \(T)\) Cartier divisors by a set of points which is called a Cartier set of \(\Sigma\). It is proved that these special divisors always exist and metric properties of the points in the corresponding Cartier set are used in order to develop the formula. An example of two combinatorially equivalent fans \(\Sigma_ 1\) and \(\Sigma_ 2\) is given such that \(X_{\Sigma_ 1}\) is projective with a non-trivial Picard group and \(X_{\Sigma_ 2}\) is non-projective with a surprisingly trivial Picard group. Furthermore, it is shown that \(\Sigma_ 2\) is a fan which cannot be spanned by any topological sphere which is the union of \((d-1)\)-polytopes such that the polytopes correspond exactly to the full dimensional cones of \(\Sigma_ 2\). Picard group; compact toric variety; Cartier divisors; fan M. Eikelberg, ''The Picard group of a compact toric variety,''Res. Math. 22, 509--527 (1992) (see also his paper ''Picard groups of compact toric varieties and combinatorial classes of fans,'' ibid.,23, 251--293 (1993). Toric varieties, Newton polyhedra, Okounkov bodies, Picard groups, Divisors, linear systems, invertible sheaves, Polyhedral manifolds The Picard group of a compact toric variety | 0 |
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 projective variety. A line bundle \(L\) on \(X\) is said to be \(3\)-very ample if the restriction map \(H^0(L) \to H^0(L|Z)\) is surjective for every degree \(4\) zero-dimensional subscheme \(Z\subset X\). The main result proved here is that if \(L\) is \(3\)-very ample and it has another technical assumption, then the secant variety \(\Sigma (X,L)\) of \(X\) of the image of \(X\) by the complete linear system \(|L|\) is normal. This technical assumption is satified if \(L\cong \omega _X\otimes A^{\otimes 2(n+1)}\otimes B\), \(n:= \dim (X)\), with \(A\) very ample and \(B\) nef. If \(X\) is a smooth curve of genus \(g\), it is sufficient to assume that \(\deg (L)\geq 2g+3\). She also proves the normality of the secant variety of the canonical embedding of all smooth curves with Clifford index \(\geq 3\). These are very nice results and they also fill a gap in the published proofs by other authors.
Among the papers used for the proofs or generalized by the present one or whose quotations are now fixed I mention: [\textit{A. Bertram}, J. Differ. Geom. 35, No. 2, 429--469 (1992; Zbl 0787.14014)], [\textit{J. Sidman} and \textit{P. Vermeire}, Algebra Number Theory 3, No. 4, 445--465 (2009; Zbl 1169.13304); Abel Symposia 6, 155--174 (2011; Zbl 1251.14043)], [\textit{P. Vermeire}, Compos. Math. 125, No. 3, 263--282 (2001; Zbl 1056.14016); J. Algebra 319, No. 3, 1264--1270 (2008; Zbl 1132.14027); Proc. Am. Math. Soc. 140, No. 8, 2639--2646 (2012; Zbl 1279.14068)]. secant variety; normality of secant variety; \(k\)-very ample line bundle; Hilbert scheme; positivity; vector bundle Ullery, B.: On the normality of secant varieties. Adv. Math. \textbf{288}, 631-647 (2016). arXiv:1408.0865v2 Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Plane and space curves, Questions of classical algebraic geometry On the normality of secant varieties | 0 |