Automorphism groups of $P^1$-bundles over rational surfaces

In this talk I will explain 1) how to construct a very simple moduli space (=a projective space) for the non-decomposable $P^1$-bundles with no "jumping fibres" over Hirzebruch surfaces, and 2) how to deduce the classification of all the $P^1$-bundles over rational surfaces whose automorphism group G is maximal in the sense that every $G$-equivariant birational map to another P1-bundle is necessarily a G-isomorphism. (This is a joint work with Jérémy Blanc and Andrea Fanelli.)