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

TL;DR

This paper studies the automorphism groups of certain Fano fibrations over projective lines, identifying invariant subsets and their orbit structures to aid in classifying algebraic subgroups of the Cremona group.

Contribution

It establishes the existence of invariant proper closed subsets and describes their orbit structure under the automorphism group for Fano fibrations over ^1, aiding classification efforts.

Findings

Existence of invariant proper closed subset as an orbit of a section.

Description of the intersection of the subset with fibers as an orbit of a subgroup.

Application to classification of algebraic subgroups of the Cremona group.

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 $S\subsetneq X$, invariant by the connected component of the identity ${\rm Aut}^{\circ}(X)$ 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 ${\rm Aut}^{\circ}(X)$ 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 ${\rm Aut}^{\circ}(X)$. This represents a first reduction step towards a possible classification of maximal connected algebraic subgroups of the Cremona group of rank $4$.