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Toric degenerations of weight varieties and applications
Philip Foth,Yi Hu +1 more
TLDR
In this paper, it was shown that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals.Abstract:
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.read more
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Integrable systems, toric degenerations and Okounkov bodies
Megumi Harada,Kiumars Kaveh +1 more
TL;DR: In this article, it was shown that if there exists a toric degeneration of a projective variety of dimension n, satisfying some natural hypotheses (which are satisfied in many settings), then there is a surjective continuous map from n to the special fiber n which is a symplectomorphism on an open dense subset of n.
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The equations for the moduli space of $n$ points on the line
TL;DR: In this paper, it was shown that the ideal of relations is generated by a set of quadric relations, with the single exception of the Segre cubic, which is generated in degree at most four and given an explicit description of the generators.
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Integrable systems, toric degenerations and Okounkov bodies
Megumi Harada,Kiumars Kaveh +1 more
TL;DR: In this paper, it was shown that if there exists a toric degeneration of a smooth complex projective variety X satisfying some natural hypotheses (which are satisfied in many settings), then there exists an integrable system on X in the sense of symplectic geometry.
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The Toric Geometry of Triangulated Polygons in Euclidean Space
TL;DR: In this paper, the authors give a geometric (Euclidean polygon) description of toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as "bendings of polygons".
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Matroids and Geometric Invariant Theory of torus actions on flag spaces
TL;DR: In this article, it was shown that if V λ [ μ ] ≠ 0, then one gets a well-defined map F / / T → CP dim V ∈ V ∆ [ ∆ ] − 1 by taking any basis of V ∀ 0.
References
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Book
Differential Geometry, Lie Groups, and Symmetric Spaces
TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.
Journal ArticleDOI
Gromov-Witten classes, quantum cohomology, and enumerative geometry
Maxim Kontsevich,Yu. I. Manin +1 more
TL;DR: In this article, the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry are discussed, and an axiomatic treatment of Gromov-Witten classes and their properties for Fano varieties are discussed.
Journal ArticleDOI
Gromov-Witten classes, quantum cohomology, and enumerative geometry
Maxim Kontsevich,Yu. I. Manin +1 more
TL;DR: In this paper, the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry are discussed, and an axiomatic treatment of Gromov-Witten classes and their properties for Fano varieties are discussed.
OtherDOI
Chow quotients of Grassmannians. I
TL;DR: In this article, a certain compactification of the space of projective configurations is introduced by considering limit position (in the Chow variety) of the closures of generic orbits, which differs considerably from Mumford's geometric invariant theory quotient.
Journal ArticleDOI
The symplectic geometry of polygons in Euclidean space
Michael Kapovich,John J. Millson +1 more
TL;DR: In this article, the authors studied the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space and showed that M r has a natural structure of a complex analytic space and is complex-analytically isomorphic to the weighted quotient of (S 2) n constructed by Deligne and Mostow.