Automorphisms of $\mathbb{P}^1$-bundles over rational surfaces

TL;DR

This paper classifies $P^1$-bundles over rational surfaces with maximal automorphism groups, providing a comprehensive understanding of their symmetry structures over algebraically closed fields of characteristic zero.

Contribution

It offers the first complete classification of such bundles with maximal automorphism groups over rational surfaces.

Findings

Identifies all $P^1$-bundles with maximal automorphism groups

Provides explicit descriptions over any algebraically closed field of characteristic zero

Establishes criteria for maximal automorphism groups in this context

Abstract

In this paper we provide the complete classification of $\mathbb{P}^1$-bundles over smooth projective rational surfaces whose neutral component of the automorphism group is maximal. Our results hold over any algebraically closed field of characteristic zero.