Rank one log del Pezzo surfaces of index two

Abstract

Let $S$ be a rank one log del Pezzo surface of index two and $S^{0}$ the smooth part of $S$. In this paper we determine the singularity type of $S$, in a way different from Alekseev and Nikulin [1]. Moreover, we calculate the fundamental group of $S^{0}$ and prove that $S$ contains the affine plane as a Zariski open subset if and only if $\pi _{1}(S^{0})=(1)$.