Related papers: A survey on Cox rings

Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the…

We consider the lattice of coarse structures on a set $X$ and study metrizable, locally finite and cellular coarse structures on $X$ from the lattice point of view.

We investigate the birational geometry of Deligne-Mumford stacks and define new birational invariants in this context.

Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject as well as external…

The aim is to give a geometric characterization of the finite generation of the Cox ring of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in…

Let G be a connected reductive group and G/H a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of G/H can be obtained by homogenizing certain equations…

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…

We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.

Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how…

Let $X$ be a projective K3 surface over $\mathbb C$. We prove that its Cox ring $R(X)$ has a generating set whose degrees are either classes of smooth rational curves, sums of at most three elements of the Hilbert basis of the nef cone, or…

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.

For an algebraic variety $X$ we study the behavior of algebraic morphisms from an algebraic variety to the group $\bir(X)$ of birational maps of $X$ and obtain, as application, some insight about the relationship between the so-called…

Given a surjective morphism $\pi\colon X\to Y$ of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of $\pi$ from the Cox ring of $X$. As a corollary we show that in some cases it…

The Cox ring of a so-called Mori Dream Space (MDS) is finitely generated and it is graded over the divisor class group. Hence the spectrum of the Cox ring comes with an action of an algebraic torus whose GIT quotient is the variety in…

Every algebraic variety can be regarded as a symplectic manifold being equipped with a Kahler form. Therefore it is natural to study lagrangian geometry of any algebraic variety. We present two basic constructions which can be applied to a…

Given a finitely generated free monoid $X$ and a morphism $\phi : X\to X$, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras…

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of previous work where toric surfaces of Picard number 1 were studied. In this article…

Let X be a smooth projective complex variety, of dimension 3, whose Hodge numbers h^{3,0}(X), h^{1,0}(X) both vanish. Let f: X--> X be a birational map that induces an isomorphism on (dense) open subvarieties U,V of X. Then we show that the…

We extend the notion of mex, which is central in combinatorial number theory, to an arbitrary combinatorial structure, and we prove a general theorem to determine the generating function of the objects having fixed mex. We then study this…