Divisors computing the minimal log discrepancy on a smooth surface

TL;DR

This paper investigates divisors that compute the minimal log discrepancy on smooth surfaces, focusing on their construction via weighted blow-ups and highlighting an example where such divisors do not compute log canonical thresholds.

Contribution

It provides a detailed analysis of divisors computing minimal log discrepancies on smooth surfaces and presents an example illustrating their limitations in computing log canonical thresholds.

Findings

Divisors computing minimal log discrepancy are obtained by weighted blow-ups.

An example shows these divisors may not compute log canonical thresholds.

The study clarifies the relationship between minimal log discrepancy and log canonical thresholds.

Abstract

We study a divisor computing the minimal log discrepancy on a smooth surface. Such a divisor is obtained by a weighted blow-up. There exists an example of a pair such that any divisor computing the minimal log discrepancy computes no log canonical thresholds.