Codimension theorems for complete toric varieties

Let be a complete toric variety with homogeneous coordinate ring . In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of generated by homogeneous polynomials that do not vanish simultaneously on .

     

Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Let \(\kappa \) be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing \(\kappa \). For a perverse \(\overline{\mathbb {Q}}_\ell \)-adic sheaf \(K_0\) on an abelian variety \(X_0\) over \(\kappa \), let K and X denote the base field extensions of \(K_0\) and \(X_0\) to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. \(\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0\). This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that \(\chi (X,K)=0\) implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

Vector bundles on toric varieties

Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.     

Arithmetic toric varieties

We study toric varieties over a field k that split in a Galois extension using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k‐forms of projective spaces when is cyclic, and we also study k‐forms of surfaces.     

Staggered t-structures on toric varieties

Achar has recently introduced a family of t-structures on the derived category of equivariant coherent sheaves on a G-scheme, generalizing the perverse coherent t-structures of Bezrukavnikov and Deligne. They are called staggered t-structures, and one of their points of interest is that they are more often self-dual. In this paper we investigate these t-structures on the T-equivariant derived category of a toric variety.     

Tilting sheaves on toric varieties

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove Kings conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number 2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.Mathematics Subject Classification (1991): 14F05; 14M25Partially supported by BFM2001-3584.     

Phylogenetic toric varieties on graphs

We define phylogenetic projective toric model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. Generators of the projective coordinate ring of the models of graphs with one cycle are explicitly described. The phylogenetic models of graphs with the same topological invariants are deformation-equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function.     

Degenerations of toric ideals and toric varieties

Toric degenerations of toric varieties and toric ideals are important both in theory and in applications. In this paper, we study the correspondence between degenerations of toric variety and of toric ideal when the weight admits a regular subdivision.     

Torsion of differentials on toric varieties

We introduce an invariant for semigroups with cancellation property. When the semigroup equals the set of lattice points in a rational, polyhedral cone, then this invariant describes the torsion of the differential sheaf on the associated toric variety. Finally, as an example, we present the case of two-dimensional cones (corresponding to two-dimensional cyclic quotient singularities).     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Gorenstein toric Fano varieties

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalizations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.Mathematics Subject Classification (2000): 14J45, 14M25, 52B20Acknowledgement The author would like to thank his thesis advisor Professor Victor Batyrev for posing problems, his advice and encouragement, as well as Professor Günter Ewald for giving reference to [Wir97] and Professor Klaus Altmann for the possibility of giving a talk at the FU Berlin. The author would also like to thank Professor Maximilian Kreuzer for the support with the computer package PALP, the classification data and many examples. Finally the author is grateful to the anonymous referee for corrections and many useful suggestions. The author was supported by DFG, Forschungsschwerpunkt Globale Methoden in der komplexen Geometrie. This work is part of the authors thesis.