Related papers: On toric varieties of high arithmetical rank

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…

In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface.…

We prove that up to automorphisms a line admits a unique embedding into the regular part of of a simplicial toric variety of dimension n>=4 over an algebraically closed field of characteristic zero which is smooth in codimension 2.

Toric varieties are perhaps the most accessible class of algebraic varieties. They often arise as varieties parameterized by monomials, and their structure may be completely understood through objects from geometric combinatorics. While…

We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admits a surjective morphism from…

A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an…

Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine…

We examine Li's double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double…

Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$…

I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian…

Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be…

We study the $A$-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We prove a Gale dual characterization of…

Toric subvarieties of projective space are classified up to projective automorphisms.

A horospherical variety is a normal algebraic variety where a reductive algebraic group acts with an open orbit which is a torus bundle over a flag variety. For example, toric varieties and flag varieties are horospherical. In this paper,…

We classify tube domains in $\mathbb C^{n+1}$ ($n\ge 1$) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least…

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…

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original…

We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.

The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over…

We express the F-signature of the coordinate ring of an affine toric variety as the volume of a certain polytope, generalizing a formula of Watanabe and Yoshida. We also compute the F-signature of pairs and triples of a toric singularity.