Some remarks on log surfaces

TL;DR

This paper investigates the behavior of log surfaces under the minimal model program, showing that smooth initial surfaces retain log terminal singularities, while singular ones may not, thus clarifying the impact of initial surface singularities.

Contribution

It proves that intermediate surfaces in the minimal model program starting from smooth pairs have only log terminal singularities, highlighting the importance of initial smoothness.

Findings

Intermediate surfaces have only log terminal singularities if starting from smooth pairs.

The property does not hold if the initial surface is singular.

The results extend the understanding of minimal model theory for log surfaces.

Abstract

Fujino and Tanaka established the minimal model theory for $\mathbb Q$-factorial log surfaces in characteristic $0$ and $p$, respectively. We prove that every intermediate surface has only log terminal singularities if we run the minimal model program starting with a pair consisting of a smooth surface and a boundary $\mathbb R$-divisor. We further show that such a property does not hold if the initial surface is singular.