Iteration of Cox rings of klt singularities

Abstract

Given a klt singularity (X, ∆; x), we define the iteration of Cox rings of (X, ∆; x). The first result of this article is that the iteration of Cox rings Cox (k) (X, ∆; x) of a klt singularity stabilizes for k large enough. The second result is a boundedness one, we prove that for a n-dimensional klt singularity (X, ∆; x) the iteration of Cox rings stabilizes for k ≥ c(n), where c(n) only depends on n. Then, we use Cox rings to establish the existence of a simply connected factorial canonical cover (or scfc cover) of a klt singularity. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano type morphisms.

Reductive quotients of klt singularities

“…However, a klt type singularity may not be factorial. In order to fix this, we will lift the action of G to the iteration of Cox rings of the singularity [13]. The lifting to the iteration of Cox rings is explained in Section 5.…”

“…In this subsection, we recall the necessary preliminaries of Cox rings and related notions. Depending on the generality and the precise setting, we refer to [6,13,33]. We recall from [6, Def 1.3.1.1] the following basic definition.…”

“…So for example in the case of local rings of singularities as considered in [13], Cox rings are not unique in general.…”

“…In the following, we will be interested in a specific choice for the subgroups of Weil divisors K and L adjusted to the local setting (cf. [13,Def. 3.15]).…”

“…For instance, we know that the fundamental group of a klt type singularity is finite [66,11] and it satisfies the Jordan property [12], while the fundamental group of a rational singularity can be an arbitrary Q-superperfect group [36]. In a similar vein, klt type singularities are local versions of Mori dream spaces [13], i.e., their local Cox ring is finitely generated, while this is no longer the case for general rational singularities. Finally, there are many interesting invariants of singularities that can be defined for klt type singularities, as the normalized volume [43] and the minimal log discrepancy [49,50].…”