Let x \subset projv be a projective spherical g variety, where v is a finite dimensional gmodule and g sp2n, c. While nonnormal toric varieties are defined in section 3. Tropical geometry in toric varieties fachbereich mathematik tu. Mar 31, 2016 this paper presents a method and a corresponding algorithm for constructing volume forms and related forms that act as kernels of integral representations on toric varieties from a convex integer polytope. We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. We introduce the notion of cracked polytope, and making use of joint work with coates and kasprzykconstruct the associated toric variety x as a subvariety of a smooth toric variety y under certain conditions. In 1989, klyachko showed that it corresponds to compatible filtrations of decreasing vector spaces 1. Given a reductive group g and a parabolic subgroup p. The software on which our experiments have been based was made public in 2009. Generalized toric varieties for simple nonrational convex. The toric variety xp of a polytope p is determined by the face fan of p, that is, the fan spanned by all faces of p. Polymake is software for the algorithmic treatment of convex polyhedra. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope.
There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. We discuss the creation of lattice polytopes defining smooth toric varieties. Equivalently, the corresponding projective variety x p has to be smooth. Little, and hal schenck the interface to this module is provided through functions. We then get a great description of the toric variety of the associahedron in terms of. These are used in chapter 6 to construct compactifications of the tropical parameter. Sagbi bases and degeneration of spherical varieties to toric. For instance, the toric variety corresponding to a simplex is a projective space. Chow stability is one notion of mumfords geometric invariant theory for studying the moduli space of polarized varieties.
We remark that the last two cases cannot be treated via standard toric geometry. A variation of this construction is to take a rational polytope in the dual of n and take the toric variety of its polar set in n. It suffices to construct an affine toric variety with this property, since it is the case that for any strongly convex cone it is possible to find a polytope of the same dimension with that cone in its normal fan. The toric variety has a map to the polytope in the dual of n whose fibers are topological tori. This toric variety does coincide with a toric ambient space obtained from a full scaffolding of a cracked polytope.
In the toric case, we can describe chow polytopes in purely combinatorial way. In this paper, we show that x can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \deltax. Lectures on toric varieties nicholas proudfoot department of mathematics, university of oregon, eugene, or 97403. The number of vertices of a fano polytope internet archive. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. If there exists a smooth projective toric variety over a simple 3dimensional polytope p, then p has at least one triangular or quadrangular face.
A symplectic toric variety x, of real dimension 2n, is completely determined by its moment polytope. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to delzant d. The ams regularly puts out nice articles titled what is. This is done by taking the vertex figure of the dual cone. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. The algorithm is implemented in the maple computer algebra system. Examples include all products of projective spaces, which are mod. In particular we show that such a variety is integral, separated and normal.
Our point of view on toric varieties here, is as images of monomial maps. The toric variety corresponding to a unit cube is the segre embedding of the fold product of the projective line. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects. The hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope.
A symplectic form on a manifold m is a closed 2form on. Facets of secondary polytopes and chow stability of toric. A toric variety xp is a certain algebraic variety or, over the real or complex num bers, a di erentiable manifold with some singularities allowedmodeled on a convex polyhedron p. A polytope is bounded if there is a ball of finite radius that contains it. The polytopes and toric varieties reu group aaron wolbach. The constructed volume forms are similar to the volume forms of the fubinistudy metric on a complex projective space.
Here we give a different proof in the smooth projective case, using noncharacteristic deformation of sheaves to find twisted polytope sheaves that corepresent the stalk functors. In the last part of the chapter, we show that every toric variety contains a dense algebraic torus. Toric varieties associated to root systems pierrelouis montagard and alvaro rittatore abstract. For instance, the degree of a toric variety with respect to a nef toric divisor is n. Additionally text output is displayed showing both the sums of the chernmather volumes of all faces of each dimension in the corresponding polytope conva starting from dimension zero and the ed degree of the corresponding toric variety x a. Projectively unique polytopes and toric slack ideals.
Brick manifolds and toric varieties of brick polytopes. Toric ideals of flow polytopes san francisco state. First you have to find equations for your toric variety, and then add a random hyperplane equation. Lattice polytopes can be used to define toric varieties with an ample line. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. Toric matrix schubert varieties and their polytopes article in proceedings of the american mathematical society 14412 august 2015 with 7 reads how we measure reads. An ane toric variety v is an irreducible ane variety such that i it contains a torus t as a zariski open subset and ii the action of t on itself extends to an action t. P is q factorial some multiple of a weil divisor is cartier if and only if the polytope. One way to construct such a variety is to take a git quotient of a. In a joint paper 17 elisa prato and fiammetta battaglia generalize the notion of toric variety and associate to each nonrational simplicial polytope a kahler quasifold and compute the betti. A macaulay2 package to compute the ed degree of a projective. The slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. Let x be a complex, gorenstein, qfactorial, toric fano variety. Kapranov, sturmfels and zelevinsky detected that chow stability of polarized toric varieties is determined by its inherent \it secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \citeksz.
This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Recently, kuwagaki \citeku2 proved that the quasiembedding is a quasiequivalence, and generalized the result to toric stacks. We prove two conjectures on the maximal picard number of x in terms of its dimension and its. The fan of a rational convex polytope in n consists of the cones over its proper faces. A polytope is said to be pointed if it contains at least one vertex. Polynomial size embeddings of toric varieties from polytopes.
Download citation brick manifolds and toric varieties of brick polytopes bottsamelson varieties factor the flag variety gb into a product of cp1s with a map into gb. Some authors use the terms convex polytope and convex polyhedron interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. Let mbe a matroid on the ground set eand basis set b. Toric varieties of brick polytopes, associahedra and. The package is implemented in macaulay2 and the internal methods make use of the polyhedra package. A toric variety is an irreducible variety xsuch that 1. Cracked polytopes and fano toric complete intersections. The rational divisor class group is the tensor product of clx with q over z. The column equations in each table describes a generating set for the ideal in the homogeneous coordinate ring of the ambient variety y described in 16. Part of the reason for this is that a nonnormal toric variety need not come from a fan see example 3. Bottsamelson varieties, subword complexes and brick.
Since all of our polyhedra and polytopes will be rational, well drop the adjective. Toric matrix schubert varieties and their polytopes. Toric birational geometry and applications to lattice polytopes. Toric varieties play an important role both in symplectic and complex geometry. G, with maximal torus t, we consider the closure x of a generic torbit in the sense of dabrowskis work, and determine when x is a gorensteinfano variety. Alvise trevisan lattice polytopes and toric varieties. Toric varieties of brick polytopes, associahedra and operad structures adam keilthy school of mathematics, trinity college dublin. Geometric invariant theory and projective toric varieties. Toric variety normal variety with zariski open, dense, c. Mathematically, a fullerene is a simple 3dimensional polytope having only pentagonal and hexagonal faces. A lattice polytope pis smooth if the normal fan at all vertices is unimodular.
This module provides support for normal toric varieties, corresponding to rational polyhedral fans. The simplest slack ideals are toric and have connections to projectively unique polytopes. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of x to exist inside y. Flips were discovered as a step of moris minimal model program. Toric birational geometry and applications to lattice. Integral polytopes are prominently featured in the theory of toric varieties, where they correspond to polarized projective toric varieties. We introduce two constructions of the toric variety of a polytope and study its basic properties. The orbit cone correspondence from theorem in a trivial way generalizes to projective toric varieties and polytopes. Twisted polytope sheaves and coherentconstructible. You need to work exclusively with one or the other. The recent interest to these results arose in connection with fullerenes. All the 2dimensional delzant polytopes with 4 vertices are trapezoids with vertices.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the ndimensional space r n. Namely, the chow polytope of a toric variety xa coincides with the secondary polytope 6 seca, which is a polytope whose vertices are corresponding to regular triangulations of q see theorem 3. Toric varieties david cox john little hal schenck department of mathematics, amherst college, amherst, ma 01002 email address. Rn, one can construct a projective toric variety x p together with an ample divisor l p on it. From cracked polytopes to fano threefolds springerlink. Computing hodge numbers in m2 with equations is usually hopeless. This group is torsion free and corresponds to the picard group if the variety is nonsingular. This post is a brief explanation about global sections of toric bundles. By default, credits for external software are shown when an external. Representation of multivariate bernoulli distributions with a given set of specified moments 2018. The toric variety of the polytope is the toric variety of its fan.
Clearly p m is a lattice polytope, hence we may consider the toric variety associated to it. Ehrhartmoment polytope hilbertvariety so the code in this repository checks the canonical strip hypothesis for smooth toric fano varieties by computing the ehrhart polynomial for the duals of the fan polytopes in the databases listed on magmas database download page. Kapranov, sturmfels and zelevinsky detected that chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope 7. Sagbi bases and degeneration of spherical varieties to. Let newf j denote thenewton polytopeof f j, if f 1f n are generic, then the number of.
A download link and examples of use are given below. The main goal of this module is to support work with gorenstein weak fano toric varieties. A toric bundle on a toric variety is just a torusequivariant locally free sheaf. The software to generate polytopes pg,n is presented in dbm12. You started with a polytope, so this cone construction should give you another polytope. If there exists a smooth projective toric variety over a simple 3 dimensional polytope p, then p has at least one triangular or quadrangular face. In 2003, perling gave a general description of torsion free sheaves based on families. Algorithm for construction of volume forms on toric varieties. We say that a quasiprojective variety x is a toric variety if xis irreducible and normal and there is a dense open subset u isomorphic to a torus, such that the natural action of u on itself extends to an action on the whole of x. Let x \subset projv be a projective spherical gvariety, where v is a finite dimensional gmodule and g sp2n, c. One mistake i always do is that i use the wrong polytope when i should use the polar polytope instead.
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