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.

SAGBI bases and degeneration of spherical varieties to toric varieties

Abstract: Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.

K-theory of affine toric varieties

Abstract: This is an updated and expanded version of my preprint #68 in the K-theory server at Urbana (which was an abstract of my talk at Vechta conference on commutative algebra, 1994). In §2 two conjectures on nilpontency of the ‘monoid Frobenius action’ on the K-theory of toric cones and on stabilizations of the corresponding K-groups are stated. Both of these conjectures are higher analogues of Anderson’s conjecture and their proof would bring a rather complete understanding of Ktheory of toric varieties/semigroup rings.

On the monomial birational maps of the projective space

Abstract: We describe the group structure of monomial Cremona transformations. It follows that every element of this group is a product of quadratic monomial transformations, and geometric descriptions in terms of fans.

Scattering amplitudes and toric geometry

Abstract: In this paper we provide a first attempt towards a toric geometric interpretation of scattering amplitudes. In recent investigations it has indeed been proposed that the all-loop integrand of planar N = 4 SYM can be represented in terms of well defined finite objects called on-shell diagrams drawn on disks. Furthermore it has been shown that the physical information of on-shell diagrams is encoded in the geometry of auxiliary algebraic varieties called the totally non negative Grassmannians. In this new formulation the infinite dimensional symmetry of the theory is manifest and many results, that are quite tricky to obtain in terms of the standard Lagrangian formulation of the theory, are instead manifest. In this paper, elaborating on previous results, we provide another picture of the scattering amplitudes in terms of toric geometry. In particular we describe in detail the toric varieties associated to an on-shell diagram, how the singularities of the amplitudes are encoded in some subspaces of the toric variety, and how this picture maps onto the Grassmannian description. Eventually we discuss the action of cluster transformations on the toric varieties. The hope is to provide an alternative description of the scattering amplitudes that could contribute in the developing of this fascinating field of research.