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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 established asymptotic formulas for the number of integral points of bounded height on toric varieties, where the integral points are bounded by a fixed number of vertices.
Abstract: We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.
25 citations
TL;DR: In this article, it was shown that for n = 3, these varieties are abelian up to finite etale coverings, and this conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory.
Abstract: These varieties are conjectured to be abelian varieties up to finite etale coverings. This conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory. In particular, this is true for n=3.
25 citations
TL;DR: In this article, it is shown that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions.
Abstract: It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.
25 citations
TL;DR: In this paper, a set of generators for the Hirzebruch elliptic genera with respect to the congruence subgroup Γ 1 (l) is presented.
Abstract: Let N⊂ℝr be a lattice, and let deg:N→ℂ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N,deg) determines a function f:ℌ→ℂ that is a holomorphic modular form of weight r for the congruence subgroup Γ1(l). Moreover, by considering all possible pairs (N ,deg), we obtain a natural subring ? (l) of modular forms with respect to Γ1 (l). We construct an explicit set of generators for ? (l), and show that ? (l) is stable under the action of the Hecke operators. Finally, we relate ? (l) to the Hirzebruch elliptic genera that are modular with respect to Γ1 (l).
25 citations
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TL;DR: The C program mori.x as discussed by the authors is a package for analyzing lattice polytopes, which is part of the PALP package for analysis of lattice lattices and is used to construct three-dimensional smooth Calabi-Yau hypersurfaces in toric varieties.
Abstract: We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric varieties are given in terms of fans over the facets of reflexive lattice polytopes. The program performs crepant star triangulations of reflexive polytopes and determines the Mori cones of the resulting toric varieties. Furthermore, it computes the intersection rings and characteristic classes of hypersurfaces.
25 citations