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 conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic surfaces was proved.
Abstract: We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic surfaces. The proof is based on a toric degeneration of cubic surfaces, the Givental's mirror theorem for toric manifolds, and the Picard-Lefschetz theory.
12 citations
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TL;DR: Rossi and Terracini as mentioned in this paper showed that every factorial complete toric variety is a finite abelian quotient of a poly weighted space (PWS), which generalizes the Batyrev-Cox and Conrads description of the Picard number.
Abstract: We prove that every \(\mathbb {Q}\)-factorial complete toric variety is a finite abelian quotient of a poly weighted space (PWS), as defined in our previous work (Rossi and Terracini in Linear Algebra Appl 495:256–288, 2016. doi:10.1016/j.laa.2016.01.039). This generalizes the Batyrev–Cox and Conrads description of a \(\mathbb {Q}\)-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) (Duke Math J 75:293–338, 1994, Lemma 2.11) and (Manuscr Math 107:215–227, 2002, Prop. 4.7), to every possible Picard number, by replacing the covering WPS with a PWS. By Buczynska’s results (2008), we get a universal picture of coverings in codimension 1 for every \(\mathbb {Q}\)-factorial complete toric variety, as topological counterpart of the \(\mathbb {Z}\)-linear universal property of the double Gale dual of a fan matrix. As a consequence, we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every \(\mathbb {Q}\)-factorial complete toric variety the description given in Rossi and Terracini (2016, Thm. 2.9) for a PWS.
12 citations
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TL;DR: In this paper, the smooth projective symmetric G-varieties with Picard number one (and G semisimple) were classified and a criterion for the smoothness of simple symmetric symmetric varieties whose closed orbit is complete.
Abstract: We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple) Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth
12 citations
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TL;DR: It is proved that the combinatorial types of those cone systems which correspond to complete smooth toric varieties are more restricted than for complete toric categories, and that the torics corresponding to essentially all types of cyclic polytopes possess singularities.
Abstract: We prove that the combinatorial types of those cone systems which correspond to complete smooth toric varieties are more restricted than for complete toric varieties: the toric varieties corresponding to essentially all types of cyclic polytopes possess singularities. This yields a negative answer to a problem stated by G. Ewald. Some consequences and problems concerning mathematical programming and the rational cohomology of smooth toric varieties are discussed.
12 citations
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TL;DR: In this article, the relation between these vertex algebras for mirror Calabi-Yau manifolds and complete intersections in toric varieties is established, which can be used to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algesbras.
Abstract: Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties. This paper could also be of interest to physicists, because it contains explicit descriptions of A and B models of Calabi-Yau hypersurfaces and complete intersection in terms of free bosons and fermions.
12 citations