Connected algebraic subgroups not lying in a maximal one

TL;DR

The paper constructs specific algebraic varieties demonstrating that their birational automorphism groups contain connected subgroups not contained within any maximal subgroup, revealing new structural insights.

Contribution

It introduces examples of ruled varieties with non-maximal connected algebraic subgroups in their birational automorphism groups, challenging previous assumptions.

Findings

Existence of ruled varieties with non-maximal connected subgroups in Bir(X)

Connected algebraic subgroups not lying in any maximal subgroup

New structural properties of birational automorphism groups

Abstract

We prove that for each $n \geq 2$, there exists a ruled variety X of dimension n such that Bir(X) contains connected algebraic subgroups which are not lying in a maximal one.