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Showing papers on "Toric variety published in 2009"


Journal ArticleDOI
TL;DR: The basic theory of toric dynamical systems is developed in the context of computational algebraic geometry and it is shown that the associated moduli space is a toric variety, which has a unique point within each invariant polyhedron.

242 citations


Journal ArticleDOI
TL;DR: The main result of as mentioned in this paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold.
Abstract: The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist.

172 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian, denoted (Gr k,n )?0.
Abstract: In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr k,n )?0. This is a cell complex whose cells Δ G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety X G . We use our technology to prove that the cell decomposition of (Gr k,n )?0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr k,n )?0 is 1.

118 citations


Journal ArticleDOI
TL;DR: In this paper, the authors constructed an A∞ category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of a smooth projective toric variety X, which is quasi-equivalent to the DG category of line bundles on X.
Abstract: Given a smooth projective toric variety X, we construct an A∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.

108 citations


Journal ArticleDOI
TL;DR: In this paper, a generalized Bott tower is constructed for toric manifolds and it is shown that if the top manifold has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial.
Abstract: If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P(E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.

98 citations


Journal ArticleDOI
TL;DR: In this article, a functorial normal crossing compactification of the moduli space of smooth cubic surfaces was proposed, analogous to the Grothendieck-Knudsen compactification.
Abstract: We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification \(M_{0,n}\subset\overline{M}_{0,n}\) .

89 citations


Journal ArticleDOI
TL;DR: In this article, different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds are investigated. And the Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phase.
Abstract: We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the `forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.

82 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give an introduction to tropical geometry and prove some results in tropical intersection theory and give a foundational account of intersection theory with proofs of new theorems relating it to classical intersection theory.

76 citations


Journal ArticleDOI
TL;DR: In this article, the authors use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds, and derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs.
Abstract: We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various points in the deformation space. Computing the instanton expansion at large volume in the flat coordinates on the open/closed deformation space we obtain predictions for enumerative invariants.

63 citations


Journal ArticleDOI
TL;DR: In this article, the authors constructed full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two.

63 citations


Journal ArticleDOI
TL;DR: The Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement is extended to affine and toric hyperplane arrangements and Zaslavsky’s fundamental results on the number of regions are generalized.
Abstract: We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.

Journal ArticleDOI
TL;DR: In this paper, the authors obtained 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes.
Abstract: We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.

Journal ArticleDOI
TL;DR: In this article, the authors generalize the result of Gel'fand et al. to a class of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.

Journal ArticleDOI
TL;DR: In this article, the authors present an upper bound on the number of possible fake weighted projective spaces with only terminal (or canonical) singularities for a fixed dimension, assuming that the singularities of the projective space have only terminal singularities.
Abstract: A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\lambda_0,...,\lambda_n). We see how the singularities of P(\lambda_0,...,\lambda_n) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios \lambda_j/\sum\lambda_i if we wish X to have only terminal (or canonical) singularities.

Journal ArticleDOI
TL;DR: In this article, the authors use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds, and derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs.
Abstract: We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various points in the deformation space. Computing the instanton expansion at large volume in the flat coordinates on the open/closed deformation space we obtain predictions for enumerative invariants.

Journal ArticleDOI
TL;DR: In this article, the authors studied the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope.
Abstract: We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let M g be the locus of nondegenerate curves inside the moduli space of curves of genus g ≥ 2. Then we show that dimM g = min(2g +1, 3g − 3), except for g = 7 where dimM 7 = 16; thus, a generic curve of genus g is nondegenerate if and only if g ≤ 4. Subject classification: 14M25, 14H10 Let k be a perfect field with algebraic closure k. Let f ∈ k[x, y] be an irreducible Laurent polynomial, and write f = ∑ (i,j)∈Z2 cijx y . We denote by supp(f) = {(i, j) ∈ Z : cij 6= 0} the support of f , and we associate to f its Newton polytope ∆ = ∆(f), the convex hull of supp(f) in R. We assume throughout that ∆ is 2-dimensional. For a face τ ⊂ ∆, let f |τ = ∑ (i,j)∈τ cijx y . We say that f is nondegenerate if, for every face τ ⊂ ∆ (of any dimension), the system of equations (1) f |τ = x ∂f |τ ∂x = y ∂f |τ ∂y = 0 has no solutions in k ∗2 . From the perspective of toric varieties, the condition of nondegeneracy can be rephrased as follows. The Laurent polynomial f defines a curve U(f) in the torus Tk = Spec k[x , y], and Tk embeds canonically in the projective toric surface X(∆)k associated to ∆ over k. Let V (f) be the Zariski closure of the curve U(f) inside X(∆)k. Then f is nondegenerate if and only if for every face τ ⊂ ∆, we have that V (f)∩Tτ is smooth of codimension 1 in Tτ , where Tτ is the toric component of X(∆)k associated to τ . (See Proposition 1.2 for alternative characterizations.) Nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry [4]: a wealth of geometric information about V (f) is contained in the combinatorics of the Newton polytope ∆(f). The notion was initially employed by Kouchnirenko [22], who studied nondegenerate polynomials in the context of singularity theory. Nondenegerate polynomials emerge naturally in the theory of sparse resultants [14] and admit a linear effective Nullstellensatz [8, Section 2.3]. They make an appearance in the study of real algebraic curves in maximal position [26] and in the problem of enumerating curves through a set of prescribed points [27]. In the case where k is a finite field, they arise in the construction of curves with many points [6, 23], in the p-adic cohomology theory of Adolphson and Sperber [2], and in explicit methods for computing zeta functions of varieties over k [8]. Despite their utility and seeming Date: 26 December 2008.

Journal ArticleDOI
TL;DR: Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus, and they can be extended to generalized toric codes as mentioned in this paper, which consists of evaluating elements of an arbitrary polynomial algebra at the torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope.

Journal ArticleDOI
TL;DR: In this article, the authors studied algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds and provided an easily verifiable sufficient condition for these properties independent of the symplectic form.
Abstract: We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.

Journal ArticleDOI
Isamu Iwanari1
TL;DR: In this paper, it was shown that there is an equivalence between the 2-category of smooth Deligne-Mumford stacks with torus embeddings and actions and the 1-categories of stacky fans.
Abstract: In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

Journal ArticleDOI
TL;DR: In this article, the authors consider reductive groups on a variety with finitely generated Cox ring and construct all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety.

Posted Content
TL;DR: In this paper, the authors show that the p1 model is a toric model specified by a multi-homogeneous ideal and conduct an extensive study of the Markov bases for p1 models that incorporate explicitly the constraint arising from multi- homogeneity.
Abstract: The p1 model is a directed random graph model used to describe dyadic interactions in a social network in terms of e!ects due to di!erential attraction (popularity) and expansiveness, as well as an additional e!ect due to reciprocation. In this article we carry out an algebraic statistics analysis of this model. We show that the p1 model is a toric model specified by a multi-homogeneous ideal. We conduct an extensive study of the Markov bases for p1 models that incorporate explicitly the constraint arising from multi- homogeneity. We consider the properties of the corresponding toric variety and relate them to the conditions for existence of the maximum likelihood and extended maximum likelihood estimator. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p1 models.

Posted Content
TL;DR: In this article, the authors describe recent results of the authors and David Nadler on microlocalization, the Fukaya category, and coherent sheaves on toric varieties, and present an expository article describing recent results.
Abstract: This is an expository article describing recent results of the authors and David Nadler on microlocalization, the Fukaya category, and coherent sheaves on toric varieties. The original papers are arXiv:math/0604379, arXiv:math/0612399 and arXiv:0811.1228v1.

Journal ArticleDOI
TL;DR: In this paper, the image of the Kodaira-Spencer map is computed for rational normal varieties with codimension one torus action and homogeneous deformations of any toric variety in arbitrary degree.
Abstract: We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.

Journal ArticleDOI
TL;DR: In this article, it was shown that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley Polytope if n⩾2d+1.

Journal ArticleDOI
Isamu Iwanari1
TL;DR: The notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stack and prove some fundamental properties, was introduced in this paper.
Abstract: In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie and Kato) and study their moduli. Then by applying this we define the notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stacks and prove some fundamental properties. Also, we study the stack-theoretic analogue of toroidal embeddings.

Posted Content
TL;DR: In this paper, the authors generalized the construction of deformations of affine toric varieties of K. Altmann and their previous constructions of weak Fano toric variants to the case of arbitrary toric components by introducing the notion of Minkowski sum decompositions of polyhedral complexes.
Abstract: We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at worst Gorenstein terminal singularities, we show that our deformations span the infinitesimal space of deformations by Kodaira-Spencer map. For Fano toric varieties, we show that their deformations can be constructed in higher-dimensional Fano toric varieties related to the Batyrev-Borisov mirror symmetry construction.

Journal ArticleDOI
TL;DR: This paper shows that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection.

Posted Content
TL;DR: In this paper, the authors characterize combinatorial objects corresponding to toric and spherical embeddings with group action, and construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k-form.
Abstract: We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field $k$. Then we provide some situations where toric varieties over $k$ are classified by Galois-stable fans, and spherical embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k$-form. In the spherical setting, we offer an example of a spherical homogeneous space $X_0$ over $\mr$ of rank 2 under the action of SU(2,1) and a smooth embedding of $X_0$ whose fan is Galois-stable but which admits no $\mr$-form.

Journal ArticleDOI
TL;DR: In this paper, the authors studied geometric interpretations of connectivity, excision results, and reinterpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A 1 -homotopy theory.

Posted Content
TL;DR: In this paper, the authors introduce toric geometry and describe toric local Calabi-Yau singularities as holomorphic quotients, and explain the gauged linear sigma-model (GLSM) Kahler quotient construction.
Abstract: These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view taken in these lectures is mostly algebro-geometric but no prior knowledge of algebraic geometry is assumed. After introducing the necessary mathematical definitions, we discuss the construction of toric varieties as holomorphic quotients. We discuss the resolution and deformation of toric Calabi-Yau singularities. We also explain the gauged linear sigma-model (GLSM) Kahler quotient construction.