Algebraic growth of the Cremona group

We initiate the study of the “algebraic growth” of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns Bir(P2), the Cremona group in 2 variables. This group is the union, for all degrees d≥1, of the algebraic variety Bir⁢(P2)d of birational transformations of the plane of degree d. Let Nd denote the number of irreducible components of Bir(P2)d. We describe the asymptotic growth of Nd as d goes to +∞, showing that there are two constants A and B>0 such that (Formula presented) for all large enough degrees d.