Log del Pezzo surfaces of rank one containing the affine plane

Abstract

Let $X$ be a log del Pezzo surface of rank one. In [8], the first author determined the possible singularity type of $X$ when $X$ contains the affine plane as a Zariski open subset. In this paper, we prove that, if $X$ contains a non-cyclic quotient singular point and its singularity type is one of the list of [8, Appendix C], then it contains the affine plane as a Zariski open subset.