Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.

Computing Gröbner fans of toric ideals

Abstract: The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Grobner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Grobner bases Enumeration by Reverse Search) for computing the Grobner fan of a toric ideal by enumerating the edge graph of its state polytope. The key contributions are an inexpensive algorithm for local change of Grobner bases in toric ideals and the identification of a reverse search tree on the vertices of the state polytope. Using these ideas we obtain a combinatorial Grobner walk procedure for toric ideals. TiGERS has been used to compute state polytopes with over 200,000 vertices.

Moduli of stable maps in genus one and logarithmic geometry II

Abstract: This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

Toric laminations, sparse generalized characteristic polynomials, and a refinement of Hilbert's tenth problem

Abstract: This paper reexamines univariate reduction from a toric geometric point of view We begin by constructing a binomial variant of the u-resultant and then retailor the generalized characteristic polynomial to sparse polynomial systems We thus obtain a fast new algorithm for univariate reduction and a better understanding of the underlying projections As a corollary, we show that a refinement of Hilbert’s Tenth Problem is decidable in single-exponential time We also obtain interesting new algebraic identities for the sparse resultant and certain multisymmetric functions

Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes

Abstract: We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at worst Gorenstein terminal singularities, we show that our deformations span the infinitesimal space of deformations by Kodaira-Spencer map. For Fano toric varieties, we show that their deformations can be constructed in higher-dimensional Fano toric varieties related to the Batyrev-Borisov mirror symmetry construction.

Flux compactification on smooth, compact three-dimensional toric varieties

Abstract: Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off their torsion classes. We illustrate our methods with explicit examples, one of which consists of an infinite class of toric CP^1 bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classification of SCTV in dimensions less or equal to 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.