Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries

TL;DR

This paper classifies certain del Pezzo surfaces of Picard rank one with low height, focusing on their geometric properties and introducing a class with elliptic boundary descendants, advancing understanding of their structure in arbitrary characteristic.

Contribution

It provides a classification of del Pezzo surfaces of Picard rank one and height at most 2, including non-log terminal cases, and introduces a class with elliptic boundary descendants.

Findings

Classified del Pezzo surfaces of height ≤ 2.

Described non-log terminal del Pezzo surfaces.

Identified del Pezzo surfaces with elliptic boundary descendants.

Abstract

This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some $\mathbb{P}^1$-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic $0$ no example of height bigger than $4$ is known. In this article, we classify del Pezzo surfaces of Picard rank one and height at most $2$; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve.