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Showing papers on "Toric variety published in 2002"


Journal ArticleDOI
TL;DR: In this paper, a stability condition for a polarised algebraic variety is defined and a conjecture relating this to the existence of a Kahler metric of constant scalar curvature.
Abstract: We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety. This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of KahlerEinstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of

945 citations


MonographDOI
09 Apr 2002
TL;DR: Polytopes topology and combinatorics of simplicial complexes Commutative and homological algebra of Cubical complexes Cubical Complexes Toric and quasitoric manifolds Moment-angle Complexes Cohomology of moment-angle complexes as discussed by the authors.
Abstract: Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

547 citations


Journal ArticleDOI
TL;DR: In this paper, the authors prove the existence of moduli spaces of pairs (P, Θ) consisting of a projective variety P with semi-abelian group action and an ample Cartier divisor on it satisfying a few simple conditions.
Abstract: I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes. the main one of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of A g . The main irreducible component of this compactification is described by an infinite periodic analog of the secondary polytope and coincides with the toroidal compactification of A g for the second Voronoi decomposition.

274 citations


Posted Content
TL;DR: In this paper, the authors discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties, and present a solution to the self-map problem in the context of algebraic algebra.
Abstract: In this note we discuss some arithmetic and geometric questions concerning self maps of projective algebraic varieties.

164 citations


Journal ArticleDOI
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 local cohomology with respect to monomial ideals to give a characteristic-free approach to vanishing results on toric varieties. As an application, we give a proof of a strong version 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.

132 citations


Posted Content
TL;DR: In this paper, the superpotential of a certain class of N = 1 supersymmetric type II compactications with fluxes and D-branes was studied, and it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet.
Abstract: We study the superpotential of a certain class ofN = 1 supersymmetric type II compactications with fluxes andD-branes. We show that it has an important two-dimensional meaning in terms of a chiral ring of the topologically twisted theory on the world-sheet. In the open-closed string B-model, this chiral ring is isomorphic to a certain relative cohomology group V , which is the appropriate mathematical concept to deal with both the open and closed string sectors. The family of mixed Hodge structures on V then implies for the superpotential to have a certain geometric structure. This structure represents a holomorphic, N = 1 supersymmetric generalization of the well-known N = 2 special geometry. It denes an integrable connection on the topological family of open-closed B-models, and a set of special coordinates on the spaceM of vev’s inN = 1 chiral multiplets. We show that it can be given a very concrete and simple realization for linear sigma models, which leads to a powerful and systematic method for computing the exact non-perturbativeN = 1 superpotentials for a broad class of toric D-brane geometries.

127 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that every Schubert variety of G has a flat degeneration into a toric variety, which is a generalization of results of [9], [7], and [6].
Abstract: LetG be a simply connected semisimple complex algebraic group. We prove that every Schubert variety ofG has a flat degeneration into a toric variety. This provides a generalization of results of [9], [7], [6]. Our basic tool is Lusztig's canonical basis and the string parametrization of this basis.

126 citations


Journal ArticleDOI
TL;DR: PalP as discussed by the authors is a C program for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points.
Abstract: We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialised to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi-Yau varieties. The package is well tested and optimised in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.

122 citations


Journal Article
TL;DR: In this paper, the authors develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra, and give the toric quiver varieties, in the sense of Nakajima.
Abstract: Extending work of Bielawski-Dancer (3) and Konno (12), we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkahler variety is a complete intersection in a Lawrence toric variety. Both varieties are non- compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov (10), are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima (15).

96 citations


Journal ArticleDOI
TL;DR: A toric surface patch associated with a convex polygon, which has vertices with integer coordinates, is defined, which naturally generalizes classical Bézier surfaces.
Abstract: We define a toric surface patch associated with a convex polygon, which has vertices with integer coordinates. This rational surface patch naturally generalizes classical Bezier surfaces. Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points. The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.

92 citations


Posted Content
TL;DR: In this paper, a theory of toric hyperkahler varieties is developed, which involves toric geometry, matroid theory and convex polyhedra, and is extended to semi-projective toric varieties.
Abstract: Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions A toric hyperkahler variety is a complete intersection in a Lawrence toric variety Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov, are extended to the hyperkahler setting When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima

Journal ArticleDOI
TL;DR: This work treats Hirzebruch surfaces and an explicit construction of an error-correcting code of length (q-1)2 over the finite field 𝔽q, obtained by evaluation of rational functions on a toric surface associated to the polytope.
Abstract: For any integral convex polytope in ℝ there is an explicit construction of an error-correcting code of length (q-1)2 over the finite field 𝔽q, obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces.

Journal ArticleDOI
TL;DR: In this paper, a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra was generated.
Abstract: During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have also completed the four-dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the context of superstring dualities. In this survey we want to give background information both on how we obtained these data, which can be found at our web site, and on how they may be used. We give a complete exposition of our classification algorithm at a mathematical (rather than algorithmic) level. We also describe how fibration structures manifest themselves in terms of toric diagrams and how we managed to find the respective data. Both for our classification scheme and for simple descriptions of fibration structures the concept of weight systems plays an important role.

Book ChapterDOI
01 Jan 2002
TL;DR: In this paper, the geometric invariants of algebraic subtorus are studied, i.e., the G-invariant open sets, the G invariant divisors, and the g-fixed points.
Abstract: Whenever one studies an algebraic variety X on which an algebraic group G acts, one of the natural questions is which geometric quantites are preserved by the action, i.e. what are the geometric invariants? For example, one may want to know the G-invariant open sets, the G-invariant divisors or, possibly, the G-invariant points, that is, the G-fixed points. If G is an algebraic torus, its geometric invariants are often very rich, especially if X is projective. In particular, an algebraic torus acting on a projective variety always has fixed points. Therefore, one of the most fruitful techniques for studying a general G-action is to consider the induced action of an algebraic subtorus.

Journal ArticleDOI
TL;DR: The toric Hilbert scheme as discussed by the authors parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal, and parametrization of the Hilbert function can be seen as a special case of the toric scheme.
Abstract: We introduce and study the toric Hilbert scheme that parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal.

Journal ArticleDOI
TL;DR: Using toric geometry, lattice theory, and elliptic surface techniques, this paper computed the Picard lattice of certain K3 surfaces and examined the generic member of each of M Reid's list of 95 families of Gorenstein k3 surfaces which occur as hypersurfaces in weighted projective 3-spaces.
Abstract: Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces In particular, we examine the generic member of each of M Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list

Book ChapterDOI
Kenji Matsuki1
01 Jan 2002
TL;DR: In this article, the authors play around with tone varieties and see all the ingredients of the Mori program at work in terms of the concrete geometry of convex cones, following the paper Reid [5].
Abstract: This chapter is intended as a coffee break after the previous thirteen chapters of hard work. We will just play around with the tone varieties and see all the ingredients of the Mori program at work in terms of the concrete geometry of convex cones, following the paper Reid [5]. It is more of my personal note to his beautiful paper, only to “draw legs on the picture of a snake”

Posted Content
03 Dec 2002
TL;DR: This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling, and the relation between linear precision and the moment map is explained.
Abstract: This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the moment map. In particular, we explain the relation between linear precision and the moment map.

Journal ArticleDOI
TL;DR: New formulae are given for Chow forms, discriminants and resultants arising from (not necessarily normal) toric varieties of codimension 2.

Journal ArticleDOI
TL;DR: In this paper, a linear sigma model for the moduli space of open-string instantons is constructed for the case of holomorphic disc instantons, and the results in agreement with those of Aganagic, Klemm and Vafa are obtained.

Posted Content
TL;DR: In this paper, a class of error-correcting codes is associated to a toric variety associated with a fan defined over a finite field and an efficient decoding algorithm for the dual of a Goppa code is presented.
Abstract: In this note, a class of error-correcting codes is associated to a toric variety associated to a fan defined over a finite field $\fff_q$, analogous to the class of Goppa codes associated to a curve. For such a ``toric code'' satisfying certain additional conditions, we present an efficient decoding algorithm for the dual of a Goppa code. Many examples are given. For small $q$, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a $(n,k,d)=(49,11,28)$ code over $\fff_8$, which is better than any other known code listed in Brouwer's on-line tables for that $n$ and $k$.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss affine semigroups and several covering properties for them and algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different sizes of the defining binomial ideals).
Abstract: Affine semigroups—discrete analogues of convex polyhedral cones—mark the cross-roads of algebraic geometry, commutative algebra and integer programming. They constitute the combinatorial background for the theory of toric varieties, which is their main link to algebraic geometry. Initiated by the work of Demazure [17] and Kempf, Knudsen, Mumford and Saint-Donat [33] in the early 70s, toric geometry is still a very active area of research. However, the last decade has clearly witnessed the extensive study of affine semigroups from the other two perspectives. No doubt, this is due to the tremendously increased computational power in algebraic geometry, implemented through the theory of Grobner bases, and, of course, to the modern computers. In this article we overview those aspects of this development that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different “sizes” of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects—polytopal semigroups and algebras— provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting (Section 4). Our counterexample to certain covering conjectures (Section 3) was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.

Journal ArticleDOI
TL;DR: In this paper, a complete toric description of fibers, image, and flattening stratification of a toric morphism are given, and the authors apply this scheme to study the family of 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi.
Abstract: Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. The Maple codes that are employed for the computation are provided. Some directions for future work are listed in the end.

Posted Content
TL;DR: In this article, dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces are described.
Abstract: We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.

Journal ArticleDOI
TL;DR: In this paper, a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a Gel'fand-Kapranov-Zelevinski) system is given.
Abstract: We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a Gel’fand-Kapranov-Zelevinskiĭ (GKZ) hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore, we classify A-hypergeometric systems as analytic D-modules.

Journal ArticleDOI
TL;DR: In this article, the authors characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties, and construct equivariant affine conoids, a tool which extends the concept of an affine cone over a projective -variety to a more general framework.
Abstract: We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well-known embedding theorem of Sumihiro on quasiprojective -varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective -variety to a more general framework.

Journal ArticleDOI
TL;DR: In this paper, the authors proved an explicit formula for the sum of the Grothendieck residues of the form ω at all roots of the system of equations, where ω is any rational nform which is regular on (C \\ 0) outside the hypersurface P1... Pn = 0.
Abstract: We consider a system of n algebraic equations P1 = · · · = Pn = 0 in the space (C/0). It is assumed that the Newton polytopes of the equations are in a sufficiently general position with respect to one another. Let ω be any rational nform which is regular on (C \\ 0) outside the hypersurface P1 . . . Pn = 0. Formerly we have announced an explicit formula for the sum of the Grothendieck residues of the form ω at all roots of the system of equations. In the present paper this formula is proved.

Journal ArticleDOI
TL;DR: In this article, the dimension of the cohomology groups of the sheaf of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety is computed.
Abstract: The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)$ of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${\O_X}(D)$. We also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart polynomial, which generalizes the usual notion and behaves similar. Namely, there exists a generalization of the inversion law for a usual Hilbert-Erhart polynomial. Some applications of the Bott formula are discussed.

Journal ArticleDOI
TL;DR: It is shown that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal, suggesting the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.
Abstract: The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P 4 whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.

01 Jan 2002
TL;DR: The following are the notes to five lectures on toric Mori theory and Fano manifolds given during the school on Toric geometry which took place in Grenoble in Summer of 2000 as discussed by the authors.
Abstract: The following are the notes to five lectures on toric Mori theory and Fano manifolds given during the school on toric geometry which took place in Grenoble in Summer of 2000.