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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 article, it was shown that the Lagrangian of the toric diagram of the Calabi-Yau cone can be embedded in the Toric diagram for all q < p with fixed p and q ≥ p.
Abstract: Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q
68 citations
TL;DR: In this paper, the Hirzebruch surface is iteratively blown up three times, and it is shown that there exist no strongly exceptional sequences of length 7 which consist of line bundles.
Abstract: King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7 which consist of line bundles.
67 citations
TL;DR: In this paper, a tropicalization functor is defined to send closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed tropical toric varieties.
Abstract: We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as T=(R∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of T-points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of T-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.
67 citations
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TL;DR: In this article, a new definition of smooth toric DM stacks in the same spirit of toric varieties is given, which is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans.
Abstract: We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.
67 citations
TL;DR: The Hamiltonian torus action on an open dense set in the moduli space of flat SU (2) connections on a compact Riemann surface was constructed in this paper.
67 citations