Real forms of rational surfaces

A real form of complex quasiprojective variety $X$ is a real variety $X_0$ which complexification (as a scheme over $\mathbb{R}$) is isomorphic to $X$. So one can ask a natural question: given some complex variety, how to describe its real forms? In this short talk (based on the recent result of Mohamed Benzerga) we show that if a rational surface $X$ has an infinite number of non-equivalent real structures, then $X$ is a blowing up of projective plane at $r \geq 10$ points. Surprisingly, this result is connected with the question about solvability of the group of automorphisms of $X$ acting trivially on Picard lattice.