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

Abstract

We classify the pairs $(X,π)$, where $π\colon X\to S$ 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.