Polyhedral Divisors and Algebraic Torus Actions

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of mul...

This article is based on a series of lectures on toric varieties given at RIMS, Kyoto. We start by introducing toric varieties, their basic properties and later pass to more advanced topics relating mostly to combinatorics.

The rank of a $d$-dimensional polytope $P$ is defined by $F-(d+1)$, where $F$ denotes the number of facets of $P$. In this paper, We focus on the toric rings of $(0,1)$-polytopes with small rank. We study their normality, the torsionfreeness of their divisor class groups and the classification of their isomorphism classes.

The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a proof of the fact that every separated toric variety may be constructed from a certain fan in a Euclidean space. To our best knowledge, this proof differs essentially from the ones which can be found in the literature.

A T-variety is an algebraic variety X with an effective regular action of an algebraic torus T. Altmann and Hausen gave a combinatorial description of an affine T-variety X by means of polyhedral divisors. In this paper we compute the higher direct images of the structure sheaf of a desingularization of X in terms of this combinatorial data. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a partial criterion for a T-var...

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started in our paers "Polytopal linear groups" (J...

These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both the software packages MAGMA and GAP). Among other things, the package implements a desingularization procedure for affine toric varieties, constructs some error-correcting codes associated with toric ...

In the present paper, we give some sufficient conditions for $\operatorname{Cl}(\Bbbk[P])$ to be torsionfree, where $\operatorname{Cl}(\Bbbk[P])$ denote the divisor class group of the toric ring $\Bbbk[P]$ of an integral polytope $P$. We prove that $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is compressed, and $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is a $(0,1)$-polytope which has at most $\dim P+2$ facets. Moreover, we characterize the toric rings of $(0,...

As an alternative to the description of a toric variety by a fan in the lattice of one parameter subgroups, we present a new language in terms of what we call bunches -- these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the...

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth k*-surfaces, our results ...