Log terminal singularities, platonic tuples and iteration of Cox rings

TL;DR

This paper extends the understanding of log terminal singularities by exploring their Cox ring iterations in higher dimensions with torus actions, providing classifications and explicit examples of compound du Val threefold singularities.

Contribution

It generalizes the Cox ring iteration framework from surface singularities to higher-dimensional cases with torus actions, including a complete classification of compound du Val threefold singularities.

Findings

Log terminal surface singularities are quotients of factorial ones by finite solvable groups.

The Cox ring iteration corresponds to the derived series of these groups.

Complete classification of compound du Val threefold singularities and their Cox ring chains.

Abstract

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.