Automorphisms of unstable ${\mathbb P}^1$-bundles

TL;DR

This paper classifies the automorphism groups of unstable projective line bundles over varieties, revealing new examples of algebraic subgroups in birational automorphism groups that are not contained in any maximal subgroup.

Contribution

It provides a comprehensive description of automorphism groups of unstable ${ m P}^1$-bundles over varieties, especially when the base's automorphism group is maximal.

Findings

Characterization of automorphism groups for unstable ${ m P}^1$-bundles

Construction of examples of algebraic subgroups in ${ m Bir}({ m P}^4)$ not in any maximal subgroup

Conditions under which the automorphism group is maximal or not

Abstract

Let $P\to X$ be a ${\mathbb P}^1$-bundle over a variety $X$. The aim of this note is to understand all connected, algebraic groups $$ \operatorname{Aut}^\circ(P)\subset G\subset \operatorname{Bir}( X\times {\mathbb P}^1). $$ We get a quite complete answer if $\operatorname{Aut}^\circ(X)$ is a maximal, connected, algebraic subgroup of $\operatorname{Bir}(X)$, and $P$ is sufficiently unstable. This gives examples of connected, algebraic subgroups of $\operatorname{Bir}({\mathbb P}^4)$ that are not contained in any maximal one. Version 2: references updated.