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 article, the authors derive a formalism for describing equivariant sheaves over toric varieties, and connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety.
Abstract: In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we construct minimal resolutions of equivariant vector bundles of rank two on toric surfaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
61 citations
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TL;DR: In this paper, the problem of describing genus zero and genus one toric curves by parameterizations was studied. But they focused on describing curves by parameters, not by their defining equations, and they gave parameterizations by rational functions and non-archimedean elliptic functions.
Abstract: In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero case and by non-archimedean elliptic functions in the genus one case. For genus zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus one curves, show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.
61 citations
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TL;DR: In this paper, it was shown that Hol$ has the same homotopy groups as Map$ up to some (computable) dimension, and the proof uses a description of $Hol$ as a space of configurations of labelled points, where the labels lie in a partial monoid determined by the fan of $X$.
Abstract: Let $X$ be a compact toric variety. Let $Hol$ denote the space of based holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let $Map$ denote the corresponding space of continuous maps. We show that $Hol$ has the same homotopy groups as $Map$ up to some (computable) dimension. The proof uses a description of $Hol$ as a space of configurations of labelled points, where the labels lie in a partial monoid determined by the fan of $X$.
60 citations
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TL;DR: In this paper, the authors consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory.
60 citations
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TL;DR: In this article, the authors studied the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein Y(p)q manifolds.
Abstract: We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein Y(p,)q manifolds. Previously, using the localization technique, we have computed the ...
60 citations