Related papers: Divisor class groups of rational trinomial varieti…

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo…

In this work we obtain sufficient conditions for a variety with a torus action of complexity one to have a finite number of automorphism group orbits.

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a…

We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra we present is in…

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial…

In this paper we use formal group rings to construct an algebraic model of the $T$-equivariant oriented cohomology of smooth toric varieties. Then we compare our model with known results of equivariant cohomology of toric varieties to…

We compute the divisor class group and the Picard group of projective varieties with Hibi rings as homogeneous coordinate rings. These varieties are precisely the toric varieties associated to order polytopes. We use tools from the theory…

The Cox rings of del Pezzo surfaces are closely related to the Lie groups E_n. In this paper, we generalize the definition of Cox rings to G- surfaces defined by us earlier, where the Lie groups G=A_n, D_n or E_n. We show that the Cox ring…

Let $k$ be a fixed finite geometric extension of the rational function field $\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let $\mathcal{O}_F$…

Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the…

In this note we introduce Cox rings of singularities and explicitly compute them in the case of du Val singularities $\mathbb{D}_n,\mathbb{E}_6,\mathbb{E}_7$ and $\mathbb{E}_8$.

For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…

We provide a complete description of normal affine algebraic varieties over the real numbers endowed with an effective action of the real circle, that is, the real form of the complex multiplicative group whose real locus consists of the…

Categorial actions of braided tensor categories are defined and shown to be the right framework for a discussion of the categorial structure related to the group of braids in the cylinder. A Kauffman polynomial of links in the solid torus…

We study the equations of universal torsors on rational surfaces.

For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor…

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…

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$…

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This…