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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, the authors apply this combinatorial geometric technique to investigate the existence of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground.
Abstract: We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.
42 citations
TL;DR: This paper uses a sort of multivariate generalization of Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes and proves a general result showing that if there is a unimodular integer affine transformation taking one polytope P1 to a second polytopes P2, then the corresponding toric code are monomially equivalent.
Abstract: Toric codes are a class of m-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $$P \subseteq {\mathbb{R}}^m$$. As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all $$m \ge 2$$. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytope P 1 to a second polytope P 2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.
42 citations
TL;DR: For a finite abelian group G ⊂ GL (n,k), the authors describes the coherent component Yθ of the moduli space M θ of θ-stable McKay quiver representations, which admits a projective birational morphism obtained by variation of Geometric Invariant Theory quotient.
Abstract: For a finite abelian group G ⊂ GL (n,k), we describe the coherent component Yθ of the moduli space Mθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism [Formula] obtained by variation of Geometric Invariant Theory quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme HilbG that avoids the (typically highly singular) Hilbert scheme of |G|-points in [Formula]. To conclude, we describe the toric fan of Yθ and hence calculate the quiver representation corresponding to any point of Yθ.
42 citations
TL;DR: In this paper, it was shown that the number of types of toric Fano varieties with certain constraints on the singularities is finite, i.e., there are no infinite number of Fano types with singularity constraints.
Abstract: The authors prove that the number of types of toric Fano varieties with certain constraints on the singularities is finite.
42 citations
TL;DR: This work characterize in graph theoretical terms the elements of the universal Grobner basis of the toric ideal of a graph and provides a new degree bound.
41 citations