Connected algebraic groups acting on three-dimensional Mori fibrations

TL;DR

This paper investigates the structure of connected algebraic groups acting on rational threefolds with Mori fibrations, using minimal model and Sarkisov programs to classify maximal connected subgroups of birational automorphisms.

Contribution

It provides a classification of maximal connected algebraic subgroups of birational automorphisms of three-dimensional projective space, extending known results in the complex case.

Findings

Classification of maximal connected algebraic subgroups of $ ext{Bir}(\\mathbb{P}^3)$

Application of minimal model and Sarkisov programs to equivariant birational geometry

Recovery of most classification results of Hiroshi Umemura in the complex case

Abstract

We study the connected algebraic groups acting on Mori fibrations $X \to Y$ with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for these fibre spaces we consider the neutral component of their automorphism groups and study their equivariant birational geometry. This is done using, inter alia, minimal model program and Sarkisov program and allows us to determine the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of Hiroshi Umemura in the complex case.