标题： Cremona群的代数增长 显示英文标题

标题： Algebraic growth of the Cremona group

摘要： 我们开始研究自同构群和代数簇的双有理变换的“代数增长”。我们的主要结果涉及$\text{Bir}(\mathbb{P}^2)$，即 $2$变量的Cremona群。这个群是所有次数为$d\geq 1$的代数簇的并集，这些代数簇是平面度为$d$的双有理变换的集合$\text{Bir}(\mathbb{P}^2)_d$。令$N_d$表示$\text{Bir}(\mathbb{P}^2)_d$的不可约分支的数量。 我们描述了当 $d$ 趋向于 $+\infty$ 时，$N_d$ 的渐近增长，证明存在两个常数 $A$ 和 $B>0$，使得对于所有足够大的度数 $d$，有 $$ A\sqrt{\ln(d)} \leq \ln \left(\ln \left(\sum_{e\leq d} N_e \right) \right) \leq B \sqrt{\ln(d)} $$。 这种增长类型似乎相当不寻常，表明计算$\text{Bir}(\mathbb{P}^2)$的代数增长通常是一个具有挑战性的问题。 显示英文摘要

摘要： 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 $d\geq 1$, 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 $+\infty$, showing that there are two constants $A$ and $B>0$ such that $$ A\sqrt{\ln(d)} \leq \ln \left(\ln \left(\sum_{e\leq d} N_e \right) \right) \leq B \sqrt{\ln(d)} $$ 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.