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The homogeneous coordinate ring of a toric variety (1995)

by D A Cox
Venue:J. Algebraic Geom
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Phases of N = 2 theories in two dimensions

by Edward Witten - NUCL. PHYS. B , 1993
"... By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. The construction permits one to recover the known corres ..."
Abstract - Cited by 235 (1 self) - Add to MetaCart
By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. The construction permits one to recover the known correspondence between these types of models and to greatly extend it to include new classes of manifolds and also to include models with (0, 2) world-sheet supersymmetry. The construction also predicts the possibility of certain physical processes involving a change in the topology of space-time.

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

by Paul S. Aspinwall, Brian R. Greene, David R. Morrison - Nucl. Phys. B , 1994
"... We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear σ-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decom ..."
Abstract - Cited by 141 (20 self) - Add to MetaCart
We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear σ-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space — the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kähler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kähler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.
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...ways of building a toric variety V from a fan ∆, in addition to the method we have discussed to this point. For a more detailed discussion of the approach of this subsection the reader is referred to =-=[39]-=-. In this paper we will concern ourselves only with the holomorphic quotient although another method, the symplectic quotient, is also quite relevant. An n-dimensional toric variety V can be realized ...

Residues in toric varieties

by Eduardo Cattani, David Cox - Compositio Mathematica
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Abstract - Cited by 140 (9 self) - Add to MetaCart
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...s coordinate ring S of X, which is the polynomial ring S = C[x1, . . ., xn+r]. Here, each variable xi corresponds to the generator ηi and hence to a torus-invariant irreducible divisor Di of X. As in =-=[C1]-=-, we grade S by declaring that the monomial Π n+r i=1 xai i has degree [∑ n+r i=1 ai Di] ∈ An−1(X). We will let β = ∑n+r i=1 deg(xi) ∈ An−1(X) denote the sum of the degrees of the variables. As is wel...

The Orbifold Chow Ring of Toric Deligne-Mumford Stacks

by Lev A. Borisov, Linda Chen, Gregory G. Smith - JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY , 2004
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Abstract - Cited by 137 (7 self) - Add to MetaCart
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...e purpose of this section is to associate a smooth Deligne-Mumford stack to certain combinatorial data. This construction is inspired by the quotient construction for toric varieties; for example see =-=[6]-=-. Let N be a finitely generated abelian group of rank d. We write N for the lattice generated by N in the d-dimensional Q-vector space NQ := N# Z Q. The natural map N # N is denoted by b ## b. Let # b...

Gröbner geometry of Schubert polynomials

by Allen Knutson, Ezra Miller - Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torus-equivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
Abstract - Cited by 99 (15 self) - Add to MetaCart
Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torus-equivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torus-equivariant cohomology class of a component (a scheme-theoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant K-theory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in K-theory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are

Quantum cohomology rings of toric manifolds

by Victor V. Batyrev , 1993
"... We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary d-dimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties of q ..."
Abstract - Cited by 95 (2 self) - Add to MetaCart
We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary d-dimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties of quantum cohomology rings H ∗ ϕ (PΣ,C) which are supposed to be valid for quantum cohomology rings of Kähler manifolds.

Orbifold resolution by D-branes

by Michael R. Douglas, Brian R. Greene, David R. Morrison - Nucl. Phys. B , 1997
"... We study topological properties of the D-brane resolution of three-dimensional orbifold singularities, C3 /Γ, for finite abelian groups Γ. The D-brane vacuum moduli space is shown to fill out the background spacetime with Fayet–Iliopoulos parameters controlling the size of the blow-ups. This D-brane ..."
Abstract - Cited by 91 (6 self) - Add to MetaCart
We study topological properties of the D-brane resolution of three-dimensional orbifold singularities, C3 /Γ, for finite abelian groups Γ. The D-brane vacuum moduli space is shown to fill out the background spacetime with Fayet–Iliopoulos parameters controlling the size of the blow-ups. This D-brane vacuum moduli space can be classically described by a gauged linear sigma model, which is shown to be non-generic in a manner that projects out non-geometric regions in its phase diagram, as anticipated from a number of perspectives.
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...the phases are physically realized. This turns out to be the case for D-branes. 15To implement this quotient description, we introduce homogeneous coordinates pi, i = 0, . . ., q − 1 in the sense of =-=[35]-=-, associated to a set A consisting of q points in the lattice N. Taking integer linear combinations of these points defines a natural map T : Z A → N, (4.5) which we assume to be surjective. The trans...

On the Hodge structure of projective hypersurfaces in toric varieties

by Victor V. Batyrev, David A. Cox - Duke Math. J , 1994
"... The purpose of this paper is to explain one extension of the ideas of the Griffiths-Dolgachev-Steenbrink method for describing the Hodge theory of smooth (resp. quasismooth) hypersurfaces in complex projective spaces (resp. in weighted projective spaces). The main idea of this method is the represen ..."
Abstract - Cited by 82 (6 self) - Add to MetaCart
The purpose of this paper is to explain one extension of the ideas of the Griffiths-Dolgachev-Steenbrink method for describing the Hodge theory of smooth (resp. quasismooth) hypersurfaces in complex projective spaces (resp. in weighted projective spaces). The main idea of this method is the representation of the Hodge components H d−1−p,p (X) in the middle cohomology group of projective hypersurfaces X = {z ∈ P d: f(z) = 0} in P d = ProjC[z1,...,zd+1] using homogeneous components of the quotient of the polynomial ring C[z1,...,zd+1] by the ideal J(f) = 〈∂f/∂z1,...,∂f/∂zd+1〉. Basic references are [13, 14, 24, 29]. In this paper, we consider hypersurfaces X in compact d-dimensional toric varieties PΣ associated with complete rational polyhedral fan Σ of simplicial cones R d. According to the theory of toric varieties [12, 22, 9, 23], PΣ is defined by glueing together of affine toric varieties Aσ = SpecC[ˇσ ∩ Z d] (σ ∈ Σ) where ˇσ denotes the dual to σ cone. Weighted projective spaces are examples of toric varieties. M. Audin [1] first noticed that there exists another approach to the definition of the
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...U(Σ) is isomorphic to the polynomial ring S(Σ) = C[z1, . . ., zn]. The action D(Σ) on U(Σ) induces a canonical grading of the ring S(Σ) by elements of Cl(Σ), i.e., by characters of D(Σ). In the paper =-=[5]-=- of the second author, the polynomial ring S(Σ) together with the Cl(Σ)-grading is called the homogeneous coordinate ring of the toric variety PΣ. One nice feature of this ring is that a hypersurface ...

Vanishing Theorems on Toric Varieties

by Mircea Mustată , 2000
"... ..."
Abstract - Cited by 67 (0 self) - Add to MetaCart
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Multigraded Hilbert schemes

by Mark Haiman, Bernd Sturmfels - J. Algebraic Geom
"... We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, includ ..."
Abstract - Cited by 63 (3 self) - Add to MetaCart
We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer’s conjecture on equations defining Grothendieck’s classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes. 1.
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...ction of G on A n is the linearization of an action on projective space, or, equivalently, that (1, 1,...,1) ∈ M ⊥ , but this hypothesis is not needed. Our notation concerning toric varieties follows =-=[7]-=- and [13]. For compatibility with the standard toric variety setting, one should take k = C; although, in fact, the construction below makes sense for any k. In (29) we identified Spec k[x, x −1 ]with...

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