Some remarks on log surfaces

Abstract

Fujino and Tanaka established the minimal model theory for $\mathbf{Q}$-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 $\mathbf{R}$-divisor. We further show that such a property does not hold if the initial surface is singular.