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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 any two such families are birational, thus proving that mirror families are independent of the choice of degeneration, which coincides with Batyrev's in the small case.
Abstract: Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.
12 citations
TL;DR: All possible Hausdorff limits of translations of X as toric degenerations are identified using elementary methods and the geometry of the secondary fan of the vector configuration.
Abstract: © 2015 London Mathematical Society.A real irrational toric variety X is an analytic subset of the simplex associated to a finite configuration of real vectors. The positive torus acts on X by translation, and we consider limits of sequences of these translations. Our main result identifies all possible Hausdorff limits of translations of X as toric degenerations using elementary methods and the geometry of the secondary fan of the vector configuration. This generalizes work of Garcia-Puente et al., who used algebraic geometry and work of Kapranov, Sturmfels and Zelevinsky, when the vectors were integral.
12 citations
TL;DR: In this paper, it was shown that for any polytope Δ there are families of hypersurfaces with the Newton polytopes that are asymptotically maximal when λ tends to infinity.
Abstract: A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with \(\mathbb{Z}_2\) coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope Δ there are families of hypersurfaces with the Newton polytopes \((\lambda \Delta )_{\lambda \in \mathbb{N}}\) that are asymptotically maximal when λ tends to infinity. We also show that these results generalize to complete intersections.
12 citations
TL;DR: In this article, the authors define immaculacy as the absence of a cohomology on an algebraic variety if the derived version of the global section functor vanishes.
Abstract: We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism.
In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans.
The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.
12 citations
TL;DR: In this paper, the authors generalize Friedman's notion of d-semistability for spaces with normal crossings to admit smoothings with regular total space, and give a duality construction involving toroidal crossing varieties whose irreducible components are toric varieties.
12 citations