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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, it was shown that if there exists a surjective morphism from X to Y, then Y is a projective space, i.e., a smooth projective variety with Picard number one.
Abstract: Let X be a complete toric variety and let Y be a smooth projective variety with Picard number one. We prove that, if there exists a surjective morphism from X to Y, then Y is a projective space.
27 citations
TL;DR: In this paper, the authors studied smooth toric Fano varieties using primitive relations and toric Mori theory, and showed that for any irreducible invariant divisor D in a toric fano variety X, the difference of the Picard numbers of X and D is Ω(n 2 ).
Abstract: In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$.
In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.
27 citations
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TL;DR: In this article, the Ricci and Bakry-Emery-Ricci lower bound on the existence of toric conical Kahler-Einstein soliton metrics was shown to hold for any toric manifold.
Abstract: We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kahler-Einstein metrics in the Gromov-Hausdorff topology.
27 citations
TL;DR: The maximum likelihood (ML) degree of toric varieties, known as discrete exponential models in statistics, is studied, showing that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal A-determinant.
27 citations
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TL;DR: For a complete toric variety, an explicit formula for the localized equivariant Todd class in terms of the combinatorial data is given in this article, based on the Riemann-Roch theorem.
Abstract: For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.
27 citations