Connected algebraic groups acting on Fano fibrations over ${\mathbb{P}}^1$

Abstract

Let $X/{\mathbb{P}}^1$ be a Mori fibre space with general fibre of Picard rank at least two. We prove that there is a proper closed subset $S⊊X$, invariant by the connected component of the identity ${\mathrm{Aut}}^{\circ }(X)$ of the automorphism group of $X$, which is moreover the orbit of a section $s$ and whose intersection with a fibre is an orbit of the subgroup of ${\mathrm{Aut}}^{\circ }(X)$ acting trivially on ${\mathbb{P}}^1$. Such result is a tool to describe equivariant birational maps from $X/{\mathbb{P}}^1$ to other Mori fibre spaces and therefore finds its applications in the study of connected algebraic subgroups of ${\mathrm{Aut}}^{\circ }(X)$. This represents a first reduction step towards a possible classification of maximal connected algebraic subgroups of the Cremona group of rank $4$.

41 pages