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.     

Nontoric foliations by Lagrangian tori of toric Fano varieties

A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux’s example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed.     

On generators of ideals defining projective toric varieties

 For any ample line bundle L on a projective toric variety of dimension n, it is proved that the line bundle L ^⊗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 ^⊗(n−1) , then the defining ideal has generators of degree at most three. Received: 11 July 2001 / Revised version: 17 December 2001     

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.     

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.     

Fano varieties in index one Fano complete intersections

Let be a smooth complex complete intersection such that . Let f : S → X be a generically finite morphism from a smooth projective variety to X. Under some positivity assumption on the anticanonical divisor of S, if 2 ≤ dim S ≤ dim X − 2 we prove that the deformations of f are contained in a subvariety of codimension at least 2.     

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.     

Ladders on Fano varieties

We prove the existence of ladders on log Fano varieties of coindex less than 4, as an application of adjunction and nonvanishing. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 56. Algebraic Geometry-9, 1998.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

Motives of some Fano varieties

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives. The first author is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by TüBİTAK-BDP funds and Bilkent University research development funds.     

Fano varieties with many selfmaps

We study global log canonical thresholds of anticanonically embedded quasismooth weighted Fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a Kähler-Einstein metric on most of them, and to produce examples of Fano varieties with infinite discrete groups of birational automorphisms.     

Remarks on abelian surfaces in nonsingular toric Fano 4-folds

In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. In consequence, we determine the possibilities of these embeddings, except for the remaining 18 nonsingular toric Fano 4-folds. Received: 12 July 2002     

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.     

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.