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.

Beyond Aztec Castles: Toric Cascades in the $dP_3$ Quiver

Abstract: Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi-Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface $dP_3$. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables which are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the $dP_3$ brane tiling for these formulas in most cases.

The minimal model theorem for divisors of toric varieties

Abstract: The minimal model conjecture says that if a proper variety has non-negative Kodaira dimension, then it has a minimal model with abundance and if the Kodaira dimension is — oo, then it is birationally equivalent to a variety which has a fibration with the relatively ample anti-canonical divisor. In this paper, first we prove this conjecture for a ^-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of the ambient toric variety. Then we prove the main result: for such a divisor with the non-negative Kodaira dimension there is an algorithm to construct concretely a projective minimal model with abundance by means of \"puffing up\" the polytope. Introduction. Let k be an algebraically closed field of arbitrary characteristic. Varieties in this paper are all defined over k. Let X be a proper algebraic variety. A proper algebraic variety Y is called a minimal model of X, if (1) Y is birationally equivalent to X, (2) Y has at worst terminal singularities and (3) the canonical divisor Ky is nef. A minimal model Y is said to have abundance if the linear system \\mKγ\\ is base point free for sufficiently large m. The minimal model conjecture states: an arbitrary proper variety with K > 0 has a minimal model with abundance while an arbitrary proper variety with K = — oo has a birationally equivalent model Y with at worst terminal singularities and a fibration Y -> Z to a lower dimensional variety with — Ky relatively ample. The conjecture is classically known to hold in the 2-dimensional case. In the 3-dimensional case the conjecture for k = C is proved by Mori [4] and Kawamata [3], while it is not yet proved in higher dimension. As a special case of higher dimension, Batyrev [1] proved, among other results, the existence of a minimal model for a A -regular anti-canonical divisor of a Gorenstein Fano toric variety Tχ(A)< In this paper, first in Section 1 we prove the minimal model conjecture for every Aregular divisor X on a toric variety of arbitrary dimension by means of successive contractions of extremal rays and flips which are introduced by Reid [7]. By Bertini's theorem, for a field k of characteristic 0, the minimal model conjecture thus holds for a general member of a base point free linear system on a proper toric variety over k. An important point of this part is providing with a technical statement Corollary 1.17 which is used in the following sections. Then in Sections 2 and 3 we prove the main result: for a Z\\-regular divisor with K > 0, there exists an algorithm to construct concretely a projective minimal model with abundance by means of \"puffing up\" the polytope corresponding to the adjoint divisor. The advantage of 1991 Mathematics Subject Classification. Primary 14M25; Secondary 14Q15. Partially supported by the Grant-in-Aid for Scientific Research (No. 09640016), the Ministry of Education, Japan.

Potential functions via toric degenerations

Abstract: This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution.

Torified varieties and their geometries over {\mathbb{F}_1}

Abstract: This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over \({\mathbb Z}\) that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes–Consani and an object in the sense of Soule and show that both are varieties over \({\mathbb{F}_1}\) in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over \({\mathbb{F}_1}\) in the literature so far. Furthermore, we compare Connes–Consani’s geometry, Soule’s geometry and Deitmar’s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over \({\mathbb{F}_1}\) in the given categories.