Endomorphisms of singular del Pezzo surfaces

TL;DR

This paper proves that a specific singular del Pezzo surface with an E8 singularity admits no endomorphisms of degree greater than 1, extending classification methods to complex algebraic surfaces with singularities.

Contribution

The authors extend Chern number inequality methods to Deligne-Mumford stacks to resolve the open case of the E8 singularity in classifying endomorphisms of del Pezzo surfaces.

Findings

The E8 singular del Pezzo surface has no endomorphism of degree > 1.

Extension of Amerik-Rovinsky-Van de Ven method to stacks.

New approach applicable to other complex surface classification problems.

Abstract

A natural problem of algebraic dynamics is to classify the complex projective varieties that admit an endomorphism of degree greater than 1. Joshi solved the problem for all canonical del Pezzo surfaces with Picard number 1 except one, a surface with a du Val singularity of type $E_8$. The method of Bott vanishing does not resolve this case. We show here that the $E_8$ surface has no endomorphism of degree greater than 1. For the proof, we extend the method of Amerik-Rovinsky-Van de Ven, involving Chern number inequalities, from varieties to Deligne-Mumford stacks. This approach should be useful for other hard cases in the classification of varieties with endomorphisms.