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.

Toric geometry and F-theory/Heterotic Duality in Four Dimensions

Abstract: We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data in a manner related to the stability of these bundles. Finally, we study Calabi-Yau fourfolds corresponding to heterotic models with fivebranes wrapping curves in the base of the Calabi-Yau threefolds. We find evidence of a topology changing extremal transition on the fourfold side which corresponds, on the heterotic side, to fivebranes wrapping different curves in the same homology class in the base.

The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

Abstract: A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.

Desingularization of toric and binomial varieties

Abstract: We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that satisfy the normal flatness criterion of Hironaka. The results extend to more general varieties defined locally by binomial equations.

Problems and algorithms for affine semigroups

Abstract: Affine semigroups—discrete analogues of convex polyhedral cones—mark the cross-roads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [17] and Kempf, Knudsen, Mumford and Saint-Donat [33] in the early 70s, toric geometry is still a very active area of research. However, the last decade has clearly witnessed the extensive study of affine semigroups from the other two perspectives. No doubt, this is due to the tremendously increased computational power in algebraic geometry, implemented through the theory of Grobner bases, and, of course, to the modern computers. In this article we overview those aspects of this development that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different “sizes” of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects—polytopal semigroups and algebras— provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting (Section 4). Our counterexample to certain covering conjectures (Section 3) was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.

A Simple Proof of Gopakumar-Vafa Conjecture for Local Toric Calabi-Yau Manifolds

Abstract: We prove Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds. It is also proven that the local Gopakumar-Vafa invariants of a given class vanish at large genera.