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 authors give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes.
Abstract: We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.
17 citations
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TL;DR: In this paper, the F-signature of the coordinate ring of an affine toric variety is expressed as the volume of a polytope, generalizing a formula of Watanabe and Yoshida.
Abstract: We express the F-signature of the coordinate ring of an affine toric variety as the volume of a certain polytope, generalizing a formula of Watanabe and Yoshida. We also compute the F-signature of pairs and triples of a toric singularity.
17 citations
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TL;DR: In this article, it was shown that a closed symplectic manifold supports at most a finite number of toric structures, and that the product of two projective spaces of complex dimension at least two has a unique toric structure.
Abstract: This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff--Tolman concept of mass linear function. Using Timorin's description of the cohomology ring via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin's higher codimension barycenters.
17 citations
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TL;DR: The notion of monocritical toric metrized divisors was introduced in this article, where it was shown that for any generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure.
Abstract: We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor D on a proper toric variety X over a global eld K is monocritical if and only if for every generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of X. The key ingredient is the study of the v-adic modulus distribution of Ga- lois orbits of generic D-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.
17 citations
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TL;DR: In this article, the authors constructed two polyhedral lower bounds and one polyhedral upper bound for the nef cone of a projective variety Y using an embedding of Y into a toric variety.
Abstract: The nef cone of a projective variety Y is an important and often elusive invariant. In this paper, we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space Graphic to a wide class of varieties.
17 citations