Georgia Institute of Technology College of Sciences

Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? For del Pezzo surfaces $M$ of degree $d \geq 7$, we will give a complete classification of such finite subgroups. Furthermore, we will give a classification of, and a structure theorem for, all involutions in $\text{Mod}(M)$ for all del Pezzo surfaces $M$. This yields a positive solution to the Nielsen realization problem for involutions on $M$ and a connection to Bertini's classification of birational involutions of $\mathbb{CP}^2$ (up to conjugation by birational automorphisms of $\mathbb{CP}^2$).