Related papers: Maps of toric varieties in Cox coordinates
We give algorithms for the computation of the algebraic de Rham cohomology of open and closed algebraic sets inside projective space or other smooth complex toric varieties. The methods, which are based on Gr\"obner basis computations in…
We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover…
We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise…
We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack; and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack.…
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…
Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus…
Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a…
Given a morphism $F : X \rightarrow Y$ from a Mori Dream Space $X$ to a smooth Mori Dream Space $Y$ and quasicoherent sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ , we describe the inverse image of $\mathcal{G}$ by $F$ and the…
Studying toric varieties from a scheme-theoretical point of view leads to toric schemes, i.e. "toric varieties over arbitrary base rings". It is shown how the base ring affects the geometry of a toric scheme. Moreover, generalisations of…
We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to…
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…
Morphisms between schemes arising from multigraded rings are essential for understanding geometric relationships in algebraic geometry, yet a systematic theory for such maps has been lacking. In this paper, we develop a comprehensive…
Let X be a Mori dream space together with an effective torus action of complexity one. In this note, we construct a polyhedral divisor on a suitable covering of the projective line P^1 which corresponds to the affine spectrum of the Cox…
Let X be a smooth complete toric variety. We describe the Altmann-Ilten-Vollmert equivariant deformations of toric varieties in the language of Cox rings. More precisely we construct one parameters families of deformations of X, such that…
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include…
We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization…
We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the…
We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a…
This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular,…
Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on…