Endomorphisms of rank one Gorenstein del Pezzo surfaces

TL;DR

This work classifies when a rank-one Gorenstein del Pezzo surface X admits an int-amplified endomorphism, showing such maps exist precisely when X is a finite quotient of a toric surface by a group acting freely in codimension one and preserving the open torus (except for S'(E8)). The authors classify these toric quotients, use Bott vanishing together with lifting to quasi-universal covers, and deploy a Riemann–Roch framework for normal surfaces to detect obstructions, aided by computational tools. They provide explicit descriptions of toric and quotient-toric surfaces among the rank-one families and determine which admit int-amplified endomorphisms, strengthening the link between toric geometry and dynamical endomorphisms on singular rational surfaces. The results are complemented by concrete examples and a computational repository to verify Bott-vanishing obstructions across the classification.

Abstract

We prove that, in all except one case, a Gorenstein del Pezzo surface of Picard rank 1 admits an int-amplified endomorphism if and only if it is a quotient of a toric variety by a finite group which acts freely in codimension one and preserves the open torus. We classify all such quotients.

Theorems & Definitions (31)

• Example 1.1
• Theorem 1.2
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• proof
• Proposition 3.1
• proof