On the structure of open del Pezzo surfaces

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show that, up to well described exceptions in characteristics 2, 3 and 5, the smooth part of $X\setminus D$ admits an ${\mathbb{A}}^1$- or an ${\mathbb{A}}_{\ast }^1$-fibration, which extends to a ${\mathbb{P}}^1$-fibration of the minimal log resolution of $(X,D)$. In characteristic 0 this improves a well-known structure theorem of Miyanishi-Tsunoda. Within the proof, we classify rational anti-canonical curves contained in smooth loci of canonical del Pezzo surfaces of rank one.