Toric sets and orbits on toric varieties

Let D be an integer matrix. A toric set, namely the points in Kⁿ parametrized by the columns of D, and a toric variety are associated to D. The toric set is a subset of the toric variety. We describe the relation between the toric set and the toric variety, in terms of the orbits of the torus action on the toric variety. The toric set depends on the sign (+,−,0) pattern of the matrix D. Finally, we prove that any toric variety over an algebraically closed field can be expressed as a toric set, for an appropriate matrix.     

Gale duality and homogeneous toric varieties

A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,𝒜), where P is an abelian group and 𝒜 is a finite collection of elements in P such that 𝒜 generates the group P and for every a∈𝒜 the element a is contained in the semigroup generated by 𝒜?{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.     

Rationality for generic toric rings

We study generic toric rings. We prove that they are Golod rings, so the Poincaré series of the residue field is rational. We classify when such a ring is Koszul, and compute its rate. Also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described. Received August 13, 1997; in final form October 23, 1998     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

Matchings in simplicial complexes, circuits and toric varieties

Using a generalized notion of matching in a simplicial complex and circuits of vector configurations, we compute lower bounds for the minimum number of generators, the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Prime lattice ideals are toric ideals, i.e. the defining ideals of toric varieties.     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.     

Vanishing theorems on toric varieties in positive characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata–Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch–Thomsen–Lauritzen–Mehta, Musta?ǎ and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.     

Degenerations of toric ideals and toric varieties

Toric degenerations of toric varieties and toric ideals are important both in theory and in applications. In this paper, we study the correspondence between degenerations of toric variety and of toric ideal when the weight admits a regular subdivision.     

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.     

Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of A. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ.     

Equivariant primary decomposition and toric sheaves

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.     

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of $\mathrm{Aut}(X)$ contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.     

Cobordisms,Manifolds with Torus Action,and Functional Equations

The paper is devoted to applications of functional equations to well-known problems of compact torus actions on oriented smooth manifolds. These include the problem of Hirzebruch genera of complex cobordism classes that are determined by complex, almost complex, and stably complex structures on a fixed manifold. We consider actions with connected stabilizer subgroups. For each such action with isolated fixed points, we introduce rigidity functional equations. This is based on the localization theorem for equivariant Hirzebruch genera. We consider actions of maximal tori on homogeneous spaces of compact Lie groups and torus actions on toric and quasitoric manifolds. The arising class of equations contains both classical and new functional equations that play an important role in modern mathematical physics.     

Normal toric ideals of low codimension

Every normal toric ideal of codimension two is minimally generated by a Gröbner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.     

Binomial free resolutions for normal toric surfaces

We construct the minimal free resolution of the residue field over a normal toric surface.

     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

The Nash problem for a toric pair and the minimal log-discrepancy

This Note formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepancy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is ?∞.     

Minimal generators of toric ideals of graphs

Let IG be the toric ideal of a graph G. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal IG.     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.