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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: Mehta and Srinivas as discussed by the authors showed that a smooth image of a projective toric variety is a toric fiber and showed that such fiber can be lifted modulo $p 2.
Abstract: We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite etale cover, admit a Zariski-locally trivial fibration with toric fibers over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that a smooth image of a projective toric variety is a toric variety. In order to deal with an important special case, we develop a logarithmic variant of the characterization of ordinary varieties with trivial tangent bundle due to Mehta and Srinivas. Furthermore, we verify our conjecture for surfaces, Fano threefolds, and homogeneous spaces (answering a question posed by Buch-Thomsen-Lauritzen-Mehta). Our proofs are based on a comprehensive theory of Frobenius liftings together with a variety of other techniques including deformation theory of rational curves and Frobenius splittings.
12 citations
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TL;DR: In this article, the authors analyzed GKZ hypergeometric systems and applied them to study the quantum cohomology rings of Calabi-Yau manifolds and related properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi Yau hypersurface.
Abstract: We analyze GKZ(Gel'fand, Kapranov and Zelevinski) hypergeometric systems and apply them to study the quantum cohomology rings of Calabi-Yau manifolds. We will relate properties of the local solutions near the large radius limit to the intersection rings of a toric variety and of a Calabi-Yau hypersurface. (Talk presented at "Frontiers in Quantum Field Theory", Osaka, Japan, Dec.1995)
12 citations
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TL;DR: In this paper, the authors define a toric degeneration of an integrable system on a projective manifold, and prove the existence of such a degeneration on the flag manifold of type A.
Abstract: We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.
12 citations
TL;DR: In this paper, it was shown that the orbit closures for directing modules over tame algebras are normal and Cohen-Macaulay, based on degenerations to normal toric varieties.
12 citations
TL;DR: An algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials, based on algebraic criterions inherited from algebraic interpolation and toric geometry is presented.
12 citations