Divisors computing minimal log discrepancies on lc surfaces

TL;DR

This paper investigates divisors that compute minimal log discrepancies on lc surface germs, establishing conditions under which these divisors are Kollár components or potential lc places, thereby advancing the understanding of singularity invariants.

Contribution

It proves the existence of Kollár components computing minimal log discrepancies for klt surface germs and characterizes divisors computing these discrepancies when the germ is not Du Val.

Findings

Divisors computing minimal log discrepancies are Kollár components in klt surface germs.

In non-Du Val cases, such divisors are potential lc places.

Results clarify the structure of divisors related to singularity invariants on lc surfaces.

Abstract

Let $(X\ni x,B)$ be an lc surface germ. If $X\ni x$ is klt, we show that there exists a divisor computing the minimal log discrepancy of $(X\ni x,B)$ that is a Koll\'ar component of $X\ni x$. If $B\not=0$ or $X\ni x$ is not Du Val, we show that any divisor computing the minimal log discrepancy of $(X\ni x,B)$ is a potential lc place of $X\ni x$.