Which rational double points occur on del Pezzo surfaces?

TL;DR

This paper classifies all rational double point configurations on del Pezzo surfaces across all degrees and characteristics, extending classical results to positive characteristic and providing explicit equations for certain cases.

Contribution

It generalizes Du Val's classification of rational double points on del Pezzo surfaces to arbitrary characteristic and degree, including explicit equations for degree 1 cases.

Findings

Complete classification of RDP configurations on del Pezzo surfaces

Extension of classical results to positive characteristic

Explicit equations for degree 1 RDP del Pezzo surfaces

Abstract

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.