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Algebraic growth of the Cremona group

Alberto Calabri, Serge Cantat, Alex Massarenti, François Maucourant, Massimiliano Mella

TL;DR

The paper develops the notion of algebraic growth for birational automorphism groups, focusing on the plane Cremona group and the degree-stratified varieties Bir(P^2)_d. By linking irreducible components to homaloidal types via Hudson’s test and the Hudson tree, it derives precise subexponential but superpolynomial growth bounds: the quantity ln ln(∑_{e≤d} N_e) scales like a constant times √ln d, while ln ln N_d also exhibits intermediate growth. A central technical advance is the seedbed s(x) analysis, showing upper and lower bounds on how seeds proliferate along Hudson’s tree, including constructions yielding 2^s(x) descendants and careful control of degree growth. The results highlight the intricate combinatorics of homaloidal types and reveal that the algebraic growth of Bir(P^2) is rich and delicate, bridging birational geometry, combinatorics, and hyperbolic geometry, with several intriguing open questions for broader varieties. The findings provide both lower and upper bounds that frame the growth landscape and suggest further study into the distribution and densities of component types across degrees.

Abstract

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.

Algebraic growth of the Cremona group

TL;DR

The paper develops the notion of algebraic growth for birational automorphism groups, focusing on the plane Cremona group and the degree-stratified varieties Bir(P^2)_d. By linking irreducible components to homaloidal types via Hudson’s test and the Hudson tree, it derives precise subexponential but superpolynomial growth bounds: the quantity ln ln(∑_{e≤d} N_e) scales like a constant times √ln d, while ln ln N_d also exhibits intermediate growth. A central technical advance is the seedbed s(x) analysis, showing upper and lower bounds on how seeds proliferate along Hudson’s tree, including constructions yielding 2^s(x) descendants and careful control of degree growth. The results highlight the intricate combinatorics of homaloidal types and reveal that the algebraic growth of Bir(P^2) is rich and delicate, bridging birational geometry, combinatorics, and hyperbolic geometry, with several intriguing open questions for broader varieties. The findings provide both lower and upper bounds that frame the growth landscape and suggest further study into the distribution and densities of component types across degrees.

Abstract

We initiate the study of the ''algebraic growth'' of groups of automorphisms and birational transformations of algebraic varieties. Our main result concerns , the Cremona group in variables. This group is the union, for all degrees , of the algebraic variety of birational transformations of the plane of degree . Let denote the number of irreducible components of . We describe the asymptotic growth of as goes to , showing that there are two constants and such that for all large enough degrees . This growth type seems quite unusual and shows that computing the algebraic growth of is a challenging problem in general.

Paper Structure

This paper contains 40 sections, 24 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Automorphisms of the affine plane
  3. The Cremona group
  4. Algebraic growth type
  5. Preliminaries
  6. Homaloidal types
  7. The Noether equalities and inequality
  8. Hudson's test and Hudson's tree
  9. Parents
  10. The tree
  11. Children
  12. Irreducible components and the main intermediate statement
  13. Basic inequalities
  14. Seeds, $*$-seeds, and tails
  15. $*$-triples and $*$-seeds
  16. ...and 25 more sections

Key Result

Theorem A

The number $K_d$ of irreducible components of ${\sf{Aut}}(\mathbb{A}^2_\mathbf{k})_d$ satisfies the following properties:

Figures (1)

  • Figure 1: On the left, schematic representation of the distribution of ($m_1,m_2$) for the $3585742777$ homaloidal types of degree $200$, where darker indicates more homaloidal types. On the right, the distribution of ($m_1,m_3$) for the same degree. Both are included in triangles determined by the inequalities $m_1+m_2\leq d$, $m_2 \leq m_1$ and similarly for $m_3$. The blue curves indicates the lower bounds on $m_2$ and $m_3$ as functions of $m_1$ given by Lemma \ref{['low_bou_m2m3']}.

Theorems & Definitions (52)

  • Theorem A
  • Theorem B
  • Example 1.1
  • Example 2.1
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 42 more