Classification of del {P}ezzo surfaces of rank one. III. Height 3

TL;DR

This paper advances the classification of rank one del Pezzo surfaces by providing a detailed analysis of those with height 3, completing the understanding for surfaces up to height 4.

Contribution

It offers the first classification of singular del Pezzo surfaces of rank one with height 3, building on previous work for heights 1 and 2.

Findings

Classified all del Pezzo surfaces of height 3 and rank one.

Confirmed the height bound of 4 for most singular del Pezzo surfaces.

Extended the classification framework to include height 3 cases.

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 $\mathbb{P}^1$-fibration whose fiber meets the exceptional divisor $h$ times. It is expected that every singular del Pezzo surface of rank one is of height $h\leq 4$, with minor exceptions in characteristics $2$ and $3$. Having settled the case $h\leq 2$ in our previous article arXiv:2412.21174, we now give a classification in case the height equals $3$.