The Structure of Algebraic Families of Birational Transformations

TL;DR

This paper characterizes algebraic families of birational transformations of algebraic varieties, establishing foundational properties and subgroup classifications, with applications to morphisms and fibrations.

Contribution

It provides a detailed description of algebraic families of birational transformations and characterizes algebraic subgroups of Bir(X).

Findings

Morphisms to Bir(X) satisfy Chevalley type results.

Algebraic subgroups of Bir(X) are exactly the closed finite-dimensional subgroups.

Studied algebraic families preserving fibrations.

Abstract

We give a description of the algebraic families of birational transformations of an algebraic variety X. As an application, we show that the morphisms to Bir(X) given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of Bir(X) are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter, Hanamura, and Ramanujam.