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 $\text{Bir}({\mathbb{P}}^2)$, the Cremona group in $2$ variables. This group is the union, for all degrees $d\ge 1$, of the algebraic variety $\text{Bir}({\mathbb{P}}^2{)}_d$ of birational transformations of the plane of degree $d$. Let $N_d$ denote the number of irreducible components of $\text{Bir}({\mathbb{P}}^2{)}_d$. We describe the asymptotic growth of $N_d$ as $d$ goes to $+\mathrm{\infty }$, showing that there are two constants $A$ and $B>0$ such that $$A\sqrt{\ln (d)}\le \ln (\ln \left(\sum _{e\le d}{N}_e\right))\le B\sqrt{\ln (d)}$$ for all large enough degrees $d$. This growth type seems quite unusual and shows that computing the algebraic growth of $\text{Bir}({\mathbb{P}}^2)$ is a challenging problem in general.