Small quotient minimal log discrepancies

TL;DR

This paper establishes a uniform bound for minimal log discrepancies in n-dimensional toric quotient singularities, demonstrating an ascending chain condition near zero, and introduces a geometric Jordan property for automorphism groups.

Contribution

It proves the existence of a positive epsilon ensuring ACC for mld's in toric quotient singularities and establishes a geometric Jordan property for automorphism groups.

Findings

Existence of epsilon_n for ACC in toric quotient singularities

Verification of the geometric Jordan property for automorphism groups

Demonstration of ACC behavior near zero for mld's

Abstract

We prove that for each positive integer $n$ there exists a positive number $\epsilon_n$ so that $n$-dimensional toric quotient singularities satisfy the ACC for mld's on the interval $(0,\epsilon_n)$. In the course of the proof, we will show a geometric Jordan property for finite automorphism groups of affine toric varieties.