Showing papers on "Toric variety published in 2005"
TL;DR: In this paper, the authors established a formula for enumeration of curves of arbitrary genus in toric surfaces by means of certain lattice paths in the Newton polygon and proved that such curves can be counted by using the so-called tropical algebraic geometry.
Abstract: The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in [17]. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space Rn and holomorphic curves with certain piecewise-linear graphs there.
613 citations
TL;DR: In this paper, the superconformal gauge theory living on the world-volume of D3 branes probing the toric singularities with horizon the recently discovered Sasaki-Einstein manifolds Lp,q,r was presented.
Abstract: We present the superconformal gauge theory living on the world-volume of D3 branes probing the toric singularities with horizon the recently discovered Sasaki-Einstein manifolds Lp,q,r. Various checks of the identification are made by comparing the central charge and the R-charges of the chiral fields with the information that can be extracted from toric geometry. Fractional branes are also introduced and the physics of the associated duality cascade discussed.
221 citations
TL;DR: In this article, the authors conjecture a general formula for assigning R-charges and multiplicities for the chiral fields of all gauge theories living on branes at toric singularities.
Abstract: We conjecture a general formula for assigning R-charges and multiplicities for the chiral fields of all gauge theories living on branes at toric singularities. We check that the central charge and the dimensions of all the chiral fields agree with the information on volumes that can be extracted from toric geometry. We also analytically check the equivalence between the volume minimization procedure discovered in hep-th/0503183 and a-maximization, for the most general toric diagram. Our results can be considered as a very general check of the AdS/CFT correspondence, valid for all superconformal theories associated with toric singularities.
185 citations
TL;DR: In this paper, the superconformal gauge theory living on the world-volume of D3 branes probing the toric singularities with horizon the recently discovered Sasaki-Einstein manifolds L^{p,q,r}.
Abstract: We present the superconformal gauge theory living on the world-volume of D3 branes probing the toric singularities with horizon the recently discovered Sasaki-Einstein manifolds L^{p,q,r}. Various checks of the identification are made by comparing the central charge and the R-charges of the chiral fields with the information that can be extracted from toric geometry. Fractional branes are also introduced and the physics of the associated duality cascade discussed.
172 citations
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06 Oct 2005
TL;DR: In this article, the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalisation of the kernel of A, leading to an explicit positive formula for the extreme monomials of any A-discriminant.
Abstract: Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.
161 citations
TL;DR: In this paper, the flat toric degeneration of the manifold Fln of flags in Cn due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) was constructed as an explicit GIT quotient of the Grobner degeneration due to Knutson and Miller (Grobner geometry of Schubert polynomials, Ann. Math.
138 citations
TL;DR: In this paper, it was shown that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine toric variety, motivated by mirror symmetry.
Abstract: We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.
114 citations
TL;DR: In this article, the notion of a reflexive polytope which appeared in connection to mirror symmetry was introduced and generalized to non-ingular toric Fano varieties, and new classification results, bounds of invariants and conjectures concerning combinatorial and geometrical properties of reflexive Polytopes were derived.
Abstract: We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalizations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
106 citations
TL;DR: In this article, the authors start to form the foundations of a theory for toric arrangements, which may be considered as the periodic version of the notion of hyperplane arrangements, motivated by the counting formulas of integral polytopes.
Abstract: Motivated by the counting formulas of integral
polytopes, as in Brion and Vergne, and Szenes and Vergne, we start to form
the foundations of a theory for toric arrangements, which
may be considered as the periodic version of the
theory of hyperplane arrangements.
86 citations
TL;DR: The aim of this investigation is to clarify the algebraic-discrete aspects of a Hopf bifurcation in these special differential equations by applying concepts from toric geometry and convex geometry.
79 citations
TL;DR: In this article, a sequence of Lagrangian submanifolds with boundary on a level set of the Landau-Ginzburg mirror of a smooth toric variety X and an ample line bundle O(1) was constructed.
Abstract: Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^n with boundary on a level set of the Landau-Ginzburg mirror of X The corresponding Floer homology groups form a graded algebra under the cup product which is canonically isomorphic to the homogeneous coordinate ring of X
TL;DR: In this paper, the authors consider a special case of the Batyrev-Borisov construction for complete intersections in toric varieties and give a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections.
Abstract: This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersection complexes, which are affine manifolds with singularities. We show these affine manifolds are related by the discrete Legendre transform, thus showing that the Batyrev-Borisov construction is a special case of our more general construction. The description of the dual intersection complexes in terms of the combinatoricsof the setup generalises work of Haase and Zharkov in the toric hypersurface case, and similar work of Ruan. In particular, this gives a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections in toric varieties.
TL;DR: In this article, the authors give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multi-graded Hilbert polynomial.
Abstract: We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann’s regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert
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TL;DR: In this article, the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and an analog of the Chern character are explicitly calculated.
Abstract: We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced smooth toric DM stacks.
TL;DR: In this article, the authors constructed small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected and showed that these point sets can be easily put into convex position, providing examples of 5-dimensional polytopes with non-connected graph-of-triangulations.
Abstract: We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc.13:3 (2000), 611–637 not only in size, but also in that the associated toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.
TL;DR: In this article, a combinatorial, self-contained proof of the existence of a smooth equivariant compactification for an algebraic torus defined over an arbitrary field is given.
TL;DR: The connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions was studied in this paper.
Abstract: We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes.
TL;DR: In this article, the authors give a map from the set of families of arcs on a variety to a set of valuations on the rational function field of the variety and characterize a family of arcs which corresponds to a divisorial valuation by this map.
Abstract: This paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety We characterize a family of arcs which corresponds to a divisorial valuation by this map. We can see that both the Nash map and a certain McKay correspondence are the restrictions of this map. This paper also gives the affirmative answer to the Nash problem for a non-normal variety in a certain category. As a corollary, we get the affirmative answer for a non-normal toric variety.
TL;DR: In this article, the authors introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and define their associated face rings.
Abstract: We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner's face ring for abstract simplicial complexes [20] and Stanley's face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.
TL;DR: In this paper, a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity is provided, which combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings.
Abstract: We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds L^{a,b,c} is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily L^{a,b,a}, whose smallest member is the Suspended Pinch Point.
TL;DR: In this paper, a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature was shown to hold for reductive algebraic varieties.
Abstract: G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [D3] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.
TL;DR: In this article, the Cox ring of a toric variety is defined for a smooth projective variety X over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on X, condition which is assumed for all varieties considered in this paper.
01 Dec 2005
TL;DR: In this article, the authors describe all smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.
Abstract: We describe all of smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.
TL;DR: In this paper, a decomposition of a complex analytic hypersurface P into bunches of branches is presented, which characterizes the embedded topological types of the irreducible components of f = 0, and is characterized by some properties of the strict transform of P by the toric embedded resolution of 0 given by the second author.
Abstract: A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Le et al.
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TL;DR: In this article, it was shown that the ideal of relations is generated in degree at most four, and gave an explicit description of the generators for the ring of invariants for arbitrary weighting of the points.
Abstract: The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators for the ring of invariants have been known since the end of the nineteenth century, but the question of the relations has remained surprisingly open, and it was not even known that the relations have bounded degree. We show that the ideal of relations is generated in degree at most four, and give an explicit description of the generators. The result holds for arbitrary weighting of the points. If all the weights are even (e.g. in the case of equal weight for odd n), we show that the ideal of relations is generated by quadrics. The proof is by degenerating the moduli space to a toric variety, and following an enlarged set of generators through this degeneration.
TL;DR: In this article, it was shown that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis.
Abstract: Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.
TL;DR: In this article, the authors show that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkahler varieties, and they give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric HHDFs, which is in fact a consequence of a long-standing conjecture of Stanley.
Abstract: Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkahler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkahler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkahler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley.
TL;DR: In this article, it was shown that all nontrivial elements in higher K-groups of toric varieties are annihilated by iterations of the natural Frobenius type endomorphisms.
Abstract: It is shown that all nontrivial elements in higher K-groups of toric varieties are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties.
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TL;DR: In this paper, the authors studied the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and proved an asymptotic converse to Serre vanishing.
Abstract: We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.
TL;DR: The authors generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes, and prove the Toric Residual Mirror Conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties.
Abstract: We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties.