Real structures on rational surfaces and automorphisms acting trivially on Picard groups
Abstract
In this article, we prove that any complex smooth rational surface X which cannot be obtained by blowing up $\mathbb P^2_{\mathbb C}$ at $r\geq 10$ points has a finite number of real forms, owing to simple results about the group $\mathrm{Aut}^{\#}X$ of complex automorphisms of X which act trivially on the Picard group of X.