Isotopy classes of involutions of del Pezzo surfaces

TL;DR

This paper classifies involutions in the mapping class groups of del Pezzo surfaces, solves the Nielsen realization problem for these involutions, and characterizes special involutions arising from birational geometry using hyperbolic reflection groups.

Contribution

It provides a complete classification of involutions, proves the Nielsen realization for them, and characterizes key geometric involutions topologically, advancing understanding of surface symmetries.

Findings

Classification of all involutions in mapping class groups

Solution to the Nielsen realization problem for involutions

Topological characterization of special birational involutions

Abstract

Let $M_n := \mathbb{CP}^2 \# n\overline{\mathbb{CP}^2}$ for $0 \leq n \leq 8$ be the underlying smooth manifold of a degree $9-n$ del Pezzo surface. We prove three results about the mapping class group $\text{Mod}(M_n) := \pi_0(\text{Homeo}^+(M_n))$: 1. the classification of, and a structure theorem for, all involutions in $\text{Mod}(M_n)$, 2. a positive solution to the smooth Nielsen realization problem for involutions of $M_n$, and 3. a purely topological characterization of three remarkable types of involutions on certain $M_n$ coming from birational geometry: de Jonqui\'eres involutions, Geiser involutions, and Bertini involutions. One main ingredient is the theory of hyperbolic reflection groups.