相关论文: Log terminal singularities, platonic tuples and it…

We classify the canonical threefold singularities that allow an effective two-torus action. This extends classification results of Mori on terminal threefold singularities and of Ishida and Iwashita on toric canonical threefold…

We give an explicit description of the divisor class groups of rational trinomial varieties. As an application, we relate the iteration of Cox rings of any rational variety with torus action of complexity one to that of a Du Val surface.

In this article, we introduce an approach to study the fundamental group of a log terminal T-variety. As applications, we prove the simply connectedness of the spectrum of the Cox ring of a complex Fano variety, we compute the fundamental…

We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for…

We study Cox rings of normal threefolds on which SL2 acts with a dense orbit. Exploiting the method of U-invariants, we obtain combinatorial criteria for the total coordinate space and the base variety to have log terminal singularities.…

We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.

We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of…

We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of…

We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition…

Given a klt singularity $(X,\Delta;x)$, we define the iteration of Cox rings of $(X,\Delta;x)$. The first result of this article is that the iteration of Cox rings ${\rm Cox}^{(k)}(X,\Delta;x)$ of a klt singularity stabilizes for $k$ large…

The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.

We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal…

We shall investigate index 1 covers of 2-dimensional log terminal singularities. The main result is that the index 1 cover is canonical if the characteristic of the base field is different from 2 or 3. We also give some counterexamples in…

We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for…

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The…

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well…

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or…

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular…

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…