Showing papers on "Toric variety published in 1996"

Combinatorial Convexity and Algebraic Geometry

Abstract: 1 Combinatorial Convexity.- I. Convex Bodies.- 1. Convex sets.- 2. Theorems of Radon and Caratheodory.- 3. Nearest point map and supporting hyperplanes.- 4. Faces and normal cones.- 5. Support function and distance function.- 6. Polar bodies.- II. Combinatorial theory of polytopes and polyhedral sets.- 1. The boundary complex of a polyhedral set.- 2. Polar polytopes and quotient polytopes.- 3. Special types of polytopes.- 4. Linear transforms and Gale transforms.- 5. Matrix representation of transforms.- 6. Classification of polytopes.- III. Polyhedral spheres.- 1. Cell complexes.- 2. Stellar operations.- 3. The Euler and the Dehn-Sommerville equations.- 4. Schlegel diagrams, n-diagrams, and polytopality of spheres.- 5. Embedding problems.- 6. Shellings.- 7. Upper bound theorem.- IV. Minkowski sum and mixed volume.- 1. Minkowski sum.- 2. Hausdorff metric.- 3. Volume and mixed volume.- 4. Further properties of mixed volumes.- 5. Alexandrov-Fenchers inequality.- 6. Ehrhart's theorem.- 7. Zonotopes and arrangements of hyperplanes.- V. Lattice polytopes and fans.- 1. Lattice cones.- 2. Dual cones and quotient cones.- 3. Monoids.- 4. Fans.- 5. The combinatorial Picard group.- 6. Regular stellar operations.- 7. Classification problems.- 8. Fano polytopes.- 2 Algebraic Geometry.- VI. Toric varieties.- 1. Ideals and affine algebraic sets.- 2. Affine toric varieties.- 3. Toric varieties.- 4. Invariant toric subvarieties.- 5. The torus action.- 6. Toric morphisms and fibrations.- 7. Blowups and blowdowns.- 8. Resolution of singularities.- 9. Completeness and compactness.- VII. Sheaves and projective toric varieties.- 1. Sheaves and divisors.- 2. Invertible sheaves and the Picard group.- 3. Projective toric varieties.- 4. Support functions and line bundles.- 5. Chow ring.- 6. Intersection numbers. Hodge inequality.- 7. Moment map and Morse function.- 8. Classification theorems. Toric Fano varieties.- VIII. Cohomology of toric varieties.- 1. Basic concepts.- 2. Cohomology ring of a toric variety.- 3. ?ech cohomology.- 4. Cohomology of invertible sheaves.- 5. The Riemann-Roch-Hirzebruch theorem.- Summary: A Dictionary.- Appendix Comments, historical notes, further exercises, research problems, suggestions for further reading.- References.- List of Symbols.

Gkz-generalized hypergeometric systems in mirror symmetry of calabi-yau hypersurfaces

Abstract: We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Grobner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up toh 1,1=3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.

Recent developments in toric geometry

Abstract: This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry.

Smoothness, Semistability, and Toroidal Geometry

Abstract: We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and toric geometry. Another proof of the same result was discovered independently by F. Bogomolov and T. Pantev. The two proofs are similar in spirit but quite different in detail.

Equations Defining Toric Varieties

Abstract: This article will appear in the proceedings of the AMS Summer Institute in Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by which I mean the defining ideals of subvarieties of affine or projective space which are parametrized by monomials. Numerous open problems are given.

The chow rings

Abstract: The properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the rela- tions among the generators. Using this fact, we have described explic- itly the Chow ring for a Q-factorial toric variety as the Stanley-Reisner ring for the corresponding fan modulo the linear equivalence relation. In this paper, we calculate the Chow ring for 3-dimensional Q-factorial toric varieties having one bacl isolated singularity.

Toric Intersection Theory for Affine Root Counting

Abstract: Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in affine space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactification instead of always resorting to products of projective spaces.

The classification of transversal multiplicity-free group actions

Abstract: Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes. This result generalizes a theorem of Delzant concerning torus actions.

Height zeta functions of toric varieties

Abstract: We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors

Weight systems for toric calabi-yau varieties and reflexivity of newton polyhedra

Abstract: According to a recently proposed scheme for the classification of reflexive polyhedra, weight systems of a certain type play a prominent role. These weight systems are classified for the cases n = 3 and n = 4, corresponding to toric varieties with K3 and Calabi-Yau hypersurfaces, respectively. For n = 3 we find the well-known 95 weight systems corresponding to weighted ℙ3’s that allow transverse polynomials, whereas for n = 4 there are 184,026 weight systems, including the 7555 weight systems for weighted ℙ4’s. It is proven (without computer) that the Newton polyhedra corresponding to all these weight systems are reflexive.

One-parameter families containing three-dimensional toric Gorenstein singularities

Abstract: For affine toric varieties, the vector space T1 (containing the infinitesimal deformations) will be interpreted via Minkowski summands of cross cuts of the defining polyhedral cone. This result will be applied to study the deformation theory of (in particular non-isolated) three-dimensional Gorenstein singularities.

Singularities of Toric Varieties Associated with Finite Distributive Lattices

Abstract: With each finite lattice L we associate a projectively embedded scheme V(L)s as Hibi has shown, the lattice D is distributive if and only if V(D) is irreducible, in which case it is a toric variety We first apply Birkhoff's structure theorem for finite distributive lattices to show that the orbit decomposition of V(D) gives a lattice isomorphic to the lattice of contractions of the bounded poset of join-irreducibles \hat {P} of D Then we describe the singular locus of V(D) by applying some general theory of toric varieties to the fan dual to the order polytope of P: V(D) is nonsingular along an orbit closure if and only if each fibre of the corresponding contraction is a tree Finally, we examine the local rings and associated graded rings of orbit closures in V(D) This leads to a second (self-contained) proof that the singular locus is as described, and a similar combinatorial criterion for the normal link of an orbit closure to be irreducible

Picard groups of hypersurfaces in toric varieties

Abstract: We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.

Products of Cycles and the Todd Class of a Toric Variety

Abstract: The purpose of this paper is to show that the Todd class of a simplicial toric variety has a canonical expression in terms of products of torus-invariant divisors. The coefficients in this expression, which are generalizations of the classical Dedekind sum, are shown to satisfy a reciprocity relation which characterizes them uniquely. We achieve these results by giving an explicit formula for the push-forward of a product of cycles under a proper birational map of simplicial toric varieties. Since the introduction of toric varieties in the 1970s, finding formulas for their Todd class has been an interesting and important problem. This is partly due to a well-known application of the Riemann-Roch theorem which allows a formula for the Todd class of a toric variety to be translated directly into a formula for the number of lattice points in a lattice polytope (cf. [Dan]). An example of this application is contained in [Pom], where a formula for the Todd class of a toric variety in terms of Dedekind sums is used to obtain new lattice point formulas. Danilov [Dan] posed the problem of finding a formula for the Todd class of a toric variety in terms of the orbit closures under the torus action. Specifically, he asked if it is possible, given a lattice, to assign a rational number to each cone in the lattice such that given any fan in the lattice, the Todd class of the corresponding toric variety equals the sum of the orbit closures with coefficients given by these assigned rational numbers. Morelli [Mor] showed that such an assignment is indeed possible in a natural way if the coefficients, instead of being rational numbers, are allowed to take values in the field of rational functions on a Grassmannian of linear subspaces of the lattice. However, if it is required that the coefficients be rational numbers invariant under lattice automorphisms, such an assignment is clearly impossible. For example, the nonsingular cone a in 22 generated by (1, 0) and (0, 1) when subdivided by the ray through (1, 1) yields two cones a, and o2 which are both lattice equivalent to a. By additivity, a consequence of the fact that the Todd class pushes forward, we deduce that the coefficient assigned to a must equal 0, which is absurd. In this paper, we show that there is a canonical expression for the Todd class of a simplicial toric variety in terms of products of the torus invariant divisors. Furthermore, this expression is invariant under lattice automorphisms. That is, the coefficient of each product depends only on the set of rays with multiplicities

GKZ Hypergeometric Systems and Applications to Mirror Symmetry

Abstract: We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995)

The ring of global sections of multiples of a line bundle on a toric variety

Abstract: In this article we prove, in a simple way, that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

Torsion Points on an Algebraic Subset of an Affine Torus

Abstract: Work of Laurent and Sarnak, following a conjecture of Lang, shows that the number of torsion points of order n on an algebraic subset of an affine complex torus is polynomial periodic. In this paper, we find bounds on the degree and period of this number as a function of n. Some examples, including the number of n torsion points on Fermat curves, are computed to illustrate the methods.

The Chow rings for 3-dimensional toric varieties with one bad isolated singularity

Abstract: The properties of a toric variety have strong connection with the combinatorial structure of the corresponding fan and the relations among the generators. Using this fact, we have described explicitly the Chow ring for aQ-factorial toric variety as the Stanley-Reisner ring for the corresponding fan modulo the linear equivalence relation. In this paper, we calculate the Chow ring for 3-dimensionalQ-factorial toric varieties having one bad isolated singularity.

Torsion of differentials on toric varieties

Abstract: 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).

On the Complexity of Smooth Projective Toric Varieties

Abstract: In this paper we answer a question posed by V.V. Batyrev. The question asked if there exists a complete regular fan with more than quadratically many primitive collections. We construct a smooth projective toric variety associated to a complete regular fan $\Delta$ in R^d with $n$ generators where the number of primitive collections of $\Delta$ is at least exponential in $n-d$. We also exhibit the connection between the number of primitive collections of $\Delta$ and the facet complexity of the Gr\"obner fan of the associated integer program.

Dual polyhedra, mirror symmetry and ginzburg-landau orbifolds

Abstract: New geometrical features of the Ginzburg-Landau orbifolds are presented, for models with a typical type of superpotential. We show the one-to-one correspondence between some of the (a, c) states with U(1) charges (−1, 1) and the integral points on the dual polyhedra, which are useful tools for the construction of mirror manifolds. Relying on toric geometry, these states are shown to correspond to the (1, 1) forms coming from blowing-up processes. In terms of the above identification, it can be checked that the monomial-divisor mirror map for Ginzburg-Landau orbifolds, proposed by the author, is equivalent to that mirror map for Calabi-Yau manifolds obtained by the mathematicians.

On the Calabi-Yau Phase of (0,2) Models

Abstract: We study the Calabi-Yau phase of a certain class of (0,2) models. These are conjectured to be equivalent to exact (0,2) superconformal field theories which have been constructed recently. Using the methods of toric geometry we discuss in a few examples the problem of resolving the singularities of such models and calculate the Euler characteristic of the corresponding gauge bundles.

Andre-Quillen cohomology of monoid algebras

Abstract: We compute the Andre-Quillen cohomology of an affine toric variety. The best results are obtained either in the general case for the first three cohomology groups, or in the case of isolated singularities for all cohomology groups, respectively.