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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, the authors classify log del Pezzo C*-surfaces of Picard number 1 and Gorenstein index less than 4 and give explicit descriptions of some equivariant smoothings of Fano threefolds.
Abstract: Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to study Fano varieties with small torus actions. As a first result we classify log del Pezzo C*-surfaces of Picard number 1 and Gorenstein index less than 4. In further examples we show how classification might work in higher dimensions and we give explicit descriptions of some equivariant smoothings of Fano threefolds.

32 citations

Posted Content
TL;DR: In this article, it was shown that a Cardy-like limit of the superconformal index of 4d ρ = 4$ SYM accounts for the entropy function whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole.
Abstract: It has recently been claimed that a Cardy-like limit of the superconformal index of 4d $\mathcal{N}=4$ SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole. Here we study this Cardy-like limit for $\mathcal{N}=1$ toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.

32 citations

Posted Content
TL;DR: In this paper, the authors studied the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and proved that it has many geometric properties analogous to those of operational Chow theory.
Abstract: We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational K-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothendieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is A 1 - homotopy invariant. Furthermore, we show that the operational K-theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the K-theory of perfect complexes on any complete toric threefold surjects onto this group. Finally we identify the equivariant operational K-theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.

32 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the smoothness of a variety of zero-dimensional systems of points on a smooth projective algebraic surface, and proved that it is smooth for and an arbitrary using the Kodaira-Spenser rank map.
Abstract: We investigate the variety of zero-dimensional subschemes (that is, systems of points) and of given lengths and on a smooth projective algebraic surface . The variety is realized as the blowing-up of the direct product of Hilbert schemes of points along the incidence graph. It is proved that is naturally isomorphic to the variety of biflags , where . We also study the problem of the smoothness of . It is proved that is smooth for and an arbitrary using the Kodaira-Spenser rank map in the theory of determinantal varieties and also in the case when by means of a direct geometric consideration.

32 citations

Journal ArticleDOI
TL;DR: In this article, a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety is constructed, which is a birational modification of the principal component of the Abramovich-Chen-Gross--Gross-Siebert space.
Abstract: This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

32 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202346
2022112
2021107
2020107
2019100
201894