Automorphism groups of ${\mathbb{P}}^1$-bundles over geometrically ruled surfaces

Abstract

We classify the pairs $(X,\pi )$, where $\pi :X\to S$ is a ${\mathbb{P}}^1$-bundle over a non-rational geometrically ruled surface $S$ and ${\mathrm{Aut}}^{\circ }(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the group $\mathrm{Bir}(X/S)$. The results hold over any algebraically closed field of characteristic zero.

To appear in Journal of the London Mathematical Society