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On the structure of open del Pezzo surfaces (2412.07458v1)

Published 10 Dec 2024 in math.AG

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, and the boundary $D$ is a reduced, nonzero divisor on $X$. We show that, up to well described exceptions in characteristics 2, 3 and 5, the open part $X\setminus D$ admits an $\mathbb{A}1$- or $\mathbb{A}1_*$-fibration with no base points on the boundary 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.

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