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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 authors established a correspondence between simplicial fans and certain foliated compact complex manifolds called LVMB-manifolds, and showed that the basic cohomology of the foliation is generated in degree two.
Abstract: We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.

27 citations

Journal ArticleDOI
TL;DR: In this paper, the image of the Kodaira-Spencer map is computed for rational normal varieties with codimension one torus action and homogeneous deformations of any toric variety in arbitrary degree.
Abstract: We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.

27 citations

Journal ArticleDOI
TL;DR: The existence of relations between volumes and Intersection Theory in the presence of singularities is proved and an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety is proved.
Abstract: We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces We prove the existence of relations between volumes and Intersection Theory in the presence of singularities In particular, we can exhibit an average Bezout Equality for equidimensional varieties We also state an upper bound for the volume of a tube about a projective variety As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices

27 citations

Journal ArticleDOI
TL;DR: In this article, the authors give necessary and sufficient conditions for a toric set to be an affine toric variety, and show some applications in the field of affine fields.

27 citations

Journal ArticleDOI
Hal Schenck1
21 May 2004
TL;DR: In this article, the free resolution of the homogeneous coordinate ring was studied for a projective toric surface, and a simple application of Green's theorem yields good bounds for the linear syzygies of a projectively toric polytope.
Abstract: Associated to an n-dimensional integral convex polytope P is a toric variety X and divisor D, such that the integral points of P represent H 0 (O X (D)) We study the free resolution of the homogeneous coordinate ring ○+ m ∈ Z H 0 (mD) as a module over Sym(H 0 (O X (D))), It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface In particular, for a planar polytope P = H 0 (O X (D)), D satisfies Green's condition Np if ∂P contains at least p + 3 lattice points

27 citations


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