On the structure of open del Pezzo surfaces

TL;DR

This paper classifies the structure of open log del Pezzo surfaces of rank one, showing that their smooth parts generally admit certain fibrations, extending known results in characteristic zero to positive characteristics with specific exceptions.

Contribution

It extends the structure theorem of Miyanishi-Tsunoda to positive characteristics, providing a detailed classification of fibrations on open del Pezzo surfaces.

Findings

Most open log del Pezzo surfaces admit $A^1$- or $A^1_*$-fibrations.

The fibrations extend to $P^1$-fibrations on minimal log resolutions.

The classification includes exceptions in characteristics 2, 3, and 5.

Abstract

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 $\mathbb{A}^{1}_{*}$-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.