Classification of Du Val del Pezzo surfaces of Picard rank one in characteristic two and three

TL;DR

This paper classifies Du Val del Pezzo surfaces of Picard rank one in characteristics two and three, and explores properties of Frobenius split surfaces and their anti-canonical members, revealing smoothness and elliptic curve structures.

Contribution

It provides a classification of these surfaces in specific characteristics and analyzes the properties of Frobenius split surfaces and their anti-canonical divisors.

Findings

Frobenius split surfaces have smooth general anti-canonical members.

In characteristic two, anti-canonical members are ordinary elliptic curves.

Classification results are specific to characteristics two and three.

Abstract

In this paper, we classify Du Val del Pezzo surfaces of Picard rank one in characteristic two and three. We also show that if a Du Val del Pezzo surface is Frobenius split, then a general anti-canonical member is smooth. Furthermore, in characteristic two, it is an ordinary elliptic curve.