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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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TL;DR: In this article, the authors give a formula that expresses any class inH∗(X,Q) as a polynomial in divisor classes and formal q variables for anyX belonging to a certain class of toric varieties.
Abstract: 1.1. Background. Toric varieties admit a combinatorial description, which allows many invariants to be expressed in terms of combinatorial data. Batyrev [Ba2] and Morrison and Plesser [MP] describe the quantum cohomology rings of certain toric varieties, in terms of generators (divisors and formal q variables) and relations (linear relations and q-deformed monomial relations). The relations are easily obtained from the combinatorial data. Unfortunately, the relations alone do not tell us how to multiply cohomology classes in the quantum cohomology ring QH∗(X), or even how to express ordinary cohomology classes in H∗(X,Q) in terms of the given generators. In this paper, we give a formula that expresses any class inH∗(X,Q)—as a polynomial in divisor classes and formal q variables—for anyX belonging to a certain class of toric varieties. These expressions, along with the presentation of QH∗(X) via generators and relations, permit computation of any product of cohomology classes in QH∗(X). Let X be a complete toric variety of dimension n over the complex numbers (all varieties in this paper are over the complex numbers). This means X is a normal variety with an action by the algebraic torus (C∗)n and a dense equivariant embedding (C∗)n → X. By the theory of toric varieties (cf. [F]), such X are characterized by a fan ∆ of strongly convex polyhedral cones in N ⊗Z R, where N is the lattice Z. The cones are rational, that is, generated by lattice points. In particular, to every ray (1-dimensional cone) σ there is a unique generator ρ ∈ N such that σ ∩N = Z>0 · ρ. There is a one-to-one correspondence between such ray generators ρ and toric (i.e., torus-invariant) divisors of X. Given toric divisors D1, . . . , Dk, with corresponding ray generators ρ1, . . . , ρk, we have D1 ∩ · · · ∩Dk 6= ∅ if and only if ρ1, . . ., ρk span a cone in ∆. Hypotheses on X translate as follows into conditions on ∆: (i) X is nonsingular if and only if every cone is generated by a part of a Z-basis of N ; (ii) given that X is nonsingular: X is Fano (i.e., X has ample anticanonical class) if and only if the set of ray generators is strictly convex.

12 citations

Journal ArticleDOI
TL;DR: In this article, the lengths of extremal rays of Birational type for toric varieties are given. But they do not consider the effects of adjoint bundles of projective toric bundles.
Abstract: We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.

12 citations

Journal ArticleDOI
TL;DR: A method is given to describe the numerical class of a torus invariant surface on a projective toric manifold that can classify toric 2-Fano manifolds of the Picard number 2 or of dimension at most 4.
Abstract: In this paper, we give a method to describe the numerical class of a torus invariant surface on a projective toric manifold. As applications, we can classify toric 2-Fano manifolds of the Picard number 2 or of dimension at most 4.

12 citations

Journal ArticleDOI
01 Jan 1997
TL;DR: For any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisors is finitely generated.
Abstract: In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

12 citations


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