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Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this article, we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco, and Piene. We follow their approach of interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then apply techniques from Adjunction Theory and Mori Theory. 相似文献
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Alan Stapledon 《Advances in Mathematics》2011,(4):3622
Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system. 相似文献
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Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties. 相似文献
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Andrzej Weber 《Central European Journal of Mathematics》2004,2(3):478-492
We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in
an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results
concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complexIH
T
*
(X)⊗H*(T). We also describe the weight filtration inIH
*(X).
Supported by KBN 2P03A 00218 grant. I thank, Institute of Mathematics, Polish Academy of Science for hospitality. 相似文献
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J.R. Maurício Corrêa 《Bulletin des Sciences Mathématiques》2010,134(7):693
We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations. 相似文献
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Eva Maria Feichtner 《Proceedings of the American Mathematical Society》2003,131(6):1695-1704
We show that the real cohomology algebra of a compact toric variety of complex dimension is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.
10.
Chi Li 《Advances in Mathematics》2011,(6):4921
In this short note, based on the work of Wang and Zhu (2004) [8], we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds. 相似文献
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E. Javier Elizondo Paulo Lima‐Filho Frank Sottile Zach Teitler 《Mathematische Nachrichten》2014,287(2-3):216-241
We study toric varieties over a field k that split in a Galois extension using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k‐forms of projective spaces when is cyclic, and we also study k‐forms of surfaces. 相似文献
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In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number to each rational polyhedral cone in the lattice, such that for any toric variety with fan in the lattice, we have
This constitutes an improved answer to an old question of Danilov.
This constitutes an improved answer to an old question of Danilov.
In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli.
Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.
15.
Giulio Cotignoli 《代数通讯》2013,41(7):2564-2573
In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? n . 相似文献
16.
Building on our earlier work on toric residues and reduction, we give a proof of the mixed toric residue conjecture of Batyrev and Materov. We simplify and streamline our technique of tropical degenerations, which allows one to interpolate between two localization principles: one appearing in the intersection theory of toric quotients and the other in the calculus of toric residues. This quickly leads to the proof of the conjecture, which gives a closed formula for the summation of a generating series whose coefficients represent a certain naive count of the numbers of rational curves on toric complete intersection Calabi-Yau manifolds. 相似文献
17.
Tomohiro Uchiyama 《代数通讯》2017,45(11):4833-4845
Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite. 相似文献
18.
Markus Perling 《Geometriae Dedicata》2007,127(1):121-129
We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.
相似文献
19.
LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental
group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π1(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC
Δ to be aK (π, 1) space whereC
Δ is the complement of a real subspace arrangement associated to Δ. 相似文献