Abstract We initiate the study of the “algebraic growth” of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns Bir (P 2) Bir (P^2), the Cremona group in 2 variables. This group is the union, for all degrees d ≥ 1 d 1, of the algebraic variety Bir (P 2) d Bir (P^2) ₃ of birational transformations of the plane of degree 𝑑. Let N d N₃ denote the number of irreducible components of Bir (P 2) d Bir (P^2) ₃. We describe the asymptotic growth of N d N₃ as 𝑑 goes to + ∞ +, showing that there are two constants 𝐴 and B > 0 B>0 such that A ln (d) ≤ ln (ln (∑ e ≤ d N e)
Calabri et al. (2026) studied this question.