Classification of del Pezzo surfaces of rank one. III. Height 3

2025/08/19 by Palka, Karol, Pełka, Tomasz
#14J10 (Primary) 14D06 #14J45 #14R05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.13609

Abstract

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the height, defined as the minimal number h such that the minimal resolution of singularities admits a ℙ¹-fibration whose fiber meets the exceptional divisor h times. It is expected that every singular del Pezzo surface of rank one is of height h≤ 4, with minor exceptions in characteristics 2 and 3. Having settled the case h≤ 2 in our previous article arXiv:2412.21174, we now give a classification in case the height equals 3.

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