On singularity types of del Pezzo surfaces with rational double points in positive characteristic

TL;DR

This paper classifies singularity types of del Pezzo surfaces with rational double points in positive characteristic, focusing on their minimal resolutions, anti-canonical members, and lifting properties to the Witt ring.

Contribution

It provides a classification of singularity types for del Pezzo surfaces with rational double points in positive characteristic, including lifting conditions and anti-canonical member singularities.

Findings

Classification of singularity types in positive characteristic

Conditions for lifting to the Witt ring

Characterization of anti-canonical members' singularities

Abstract

In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo surface with rational double points whose anti-canonical members are all singular. As a corollary, we determine all singularity types of del Pezzo surfaces with rational double points which only appear in positive characteristic.