Isotopy classes of involutions of del Pezzo surfaces

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.