Algebraic families of birational transformations

The notion of an algebraic family of birational automorphisms was introduced in the famous 1970 paper by Demazure. This notion allows one to define Zariski topology on the group of birational automorphisms of an algebraic variety. In the case of the projective space, Blanc and Furter proved that the Cremona group does not have an ind-variety structure. However, its Zariski topology can be in a certain way «reconstructed» from countably many algebraic families. In a recent paper, Regeta, Urech and van Santen show an analogous result for any algebraic variety. I will discuss this result and some of its useful corollaries.