Related papers: On toric varieties of high arithmetical rank
We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to…
We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric…
In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\Nef(X)$ of numerically effective divisors…
We classify the subvarieties of infinite dimensional affine space that are stable under the infinite symmetric group. We determine the defining equations and point sets of these varieties as well as the containments between them.
A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…
The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is…
Hypersemitoric systems are a class of integrable systems on $4$-dimensional symplectic manifolds which only have mildly degenerate singularities and where one of the integrals induces an effective Hamiltonian $S^1$-action and is proper. We…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the…
We prove that several invariants of a possibly singular complex affine or projective variety of degree $d$ in the affine space $\mathbb{A}^{n}$, or $\mathbb{P}^n$, are bounded by a function of $d$ alone, provided $b_{1}=0$ for a resolution…
We translate the equivariant decomposition theorem (in the case of a proper morphism of toric varieties) in to the language of combinatorially defined ``shifted minimal complexes''.
This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid…
We discuss an experimental approach to open problems in toric geometry: are smooth projective toric varieties (i) projectively normal and (ii) defined by degree 2 equations? We discuss the creation of lattice polytopes defining smooth toric…
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same…
Based on the Reid-Shepherd-Barron-Tai criterion for canonical and terminal quotient singularities, we characterize canonicity and terminality of a toric variety in terms of its local class group actions. Specializing it to the Picard number…
Consider an algebraic torus of small dimension acting on an open subset of a complex vector space, or more generally on a quasiaffine variety such that a separated orbit space exists. We discuss under which conditions this orbit space is…
In this paper we construct $n$-valued maps on $k$-dimensional tori, where $n,k\geq 2$, that are not homotopic to affine $n$-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine…