The structure of algebraic families of birational transformations

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 $\mathrm{Bir}\left(X\right)$ given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of $\mathrm{Bir}\left(X\right)$ 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 [2], Hanamura [9], and Ramanujam [20].