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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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Journal ArticleDOI
TL;DR: In this article, the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables is formulated in terms of a pair consisting of a consistent dimer model and a perfect matching on it.
Abstract: We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.

34 citations

Journal ArticleDOI
Martin Ulirsch1
TL;DR: In this article, the Kajiwara-Payne tropicalization map is shown to have a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus.
Abstract: For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.

34 citations

Posted Content
TL;DR: In this paper, the authors show that the p1 model is a toric model specified by a multi-homogeneous ideal and conduct an extensive study of the Markov bases for p1 models that incorporate explicitly the constraint arising from multi- homogeneity.
Abstract: The p1 model is a directed random graph model used to describe dyadic interactions in a social network in terms of e!ects due to di!erential attraction (popularity) and expansiveness, as well as an additional e!ect due to reciprocation. In this article we carry out an algebraic statistics analysis of this model. We show that the p1 model is a toric model specified by a multi-homogeneous ideal. We conduct an extensive study of the Markov bases for p1 models that incorporate explicitly the constraint arising from multi- homogeneity. We consider the properties of the corresponding toric variety and relate them to the conditions for existence of the maximum likelihood and extended maximum likelihood estimator. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p1 models.

34 citations

Posted Content
TL;DR: In this article, a simple formula for computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given, and the multiplier module and test ideals are also treated.
Abstract: A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.

34 citations

Journal ArticleDOI
TL;DR: It is shown that the fuzzification of a projective toric variety amounts to a quantization of its toric base, and one can easily obtain large classes of fuzzy Calabi-Yau manifolds.
Abstract: We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we show that the fuzzification of a projective toric variety amounts to a quantization of its toric base.

34 citations


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