Topic
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.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this article, the authors generalize this representation to the case of an arbitrary smooth toric variety and show that a surjection O Y → L gives n + 1 sections of L which don't vanish simultaneously and hence determine a map Y → Pnk.
Abstract: (1) Y 7→ {line bundle quotients of O Y } . This is easy to prove since a surjection O Y → L gives n + 1 sections of L which don’t vanish simultaneously and hence determine a map Y → Pnk . The goal of this paper is to generalize this representation to the case of an arbitrary smooth toric variety. We will work with schemes over an algebraically closed field k of characteristic zero, and we will fix a smooth n-dimensional toric variety X determined by the fan ∆ in NR = R. As usual, M denotes the dual lattice of N and ∆(1) denotes the 1-dimensional cones of ∆. We will use ∑
93 citations
TL;DR: In this paper, the authors studied the K-theory of diagonalizable group schemes on noetherian regular separated algebraic spaces, and showed how to reconstruct the Ktheory ring of such an action from the ktheory rings of the loci where the stabilizers have constant dimension.
Abstract: We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.
92 citations
Posted Content•
TL;DR: In this paper, it was shown that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.
Abstract: We use Cox's description for sheaves on toric varieties and results about the local cohomology with respect to monomial ideals to give a characteristic free approach to vanishing results on arbitrary toric varieties. As an application, we give a proof of a strong form of Fujita's conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.
92 citations
TL;DR: In this article, the Chow ring of a smooth toric variety was constructed from a finite lattice and a subset of a subset in a building set, a so-called building set.
Abstract: We study a graded algebra \(D=D(\mathcal{L},\mathcal{G})\) over ℤ defined by a finite lattice ℒ and a subset \(\mathcal{G}\) in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and \(\mathcal{G}\). We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Grobner basis of the relation ideal of D and a monomial basis of D.
92 citations
TL;DR: In this article, the behavior of algebraic and topological invariants under morphisms of projective algebraic varieties was studied and the results were applied to obtain a version of Grothendieck-Riemann-Roch.
Abstract: This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices.
92 citations