Topic
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.
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TL;DR: In this paper, it was shown that the Dynkin diagrams of nonabelian gauge groups occurring in type IIA and F-theory can be read off from the polyhedron Δ∗ that provides the toric description of the Calabi-Yau manifold used for compactification.
60 citations
TL;DR: In this paper, Gelfand, Kapranov and Zelevinsky gave explicit formulas for the dimensions and the degrees of A-discriminant varieties introduced by GELFAND, Karpathy and Zisserman and applied them to the case of higher-codimensional toric varieties.
60 citations
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TL;DR: In this paper, a convex body is associated to an affine variety X and a finite dimensional vector space of regular functions L on X, such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes.
Abstract: Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).
60 citations
TL;DR: In this article, a rational convex polytope of dimension r>=2 was constructed from a rational function of the toric variety defined at the algebraic torus, and the minimum distance of the code was estimated using intersection theory and mixed volumes.
59 citations
TL;DR: In this article, the authors studied the geometry of cut ideals and the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal.
Abstract: Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal. Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables.
59 citations