Endomorphisms of singular del Pezzo surfaces

TL;DR

The paper tackles the problem of classifying normal complex projective varieties admitting endomorphisms of degree greater than one, focusing on canonical del Pezzo surfaces with Picard number 1 and the elusive E8 du Val singularity case. It extends the Amerik-Rovinsky-Van de Ven approach by developing Chern number inequalities for Deligne-Mumford stacks, which rely on precise global-generation properties of twisted cotangent bundles. A key result is that the E8 del Pezzo surface X with no nonzero vector fields has no endomorphism of degree greater than one and is not an image of a proper toric variety, resolving the remaining canonical Picard-1 case and completing related image-classification statements. The work provides a stack-based framework that could handle other hard cases in the broader classification of varieties with endomorphisms.

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.

Figures (1)

• Figure 1: The singular curves in the elliptic pencil $|O_M(1)|$ on $M$

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Theorem 3.1
• proof