Algebraic growth of the Cremona group
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913722504577024 |
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| author | Calabri, Alberto Cantat, Serge Massarenti, Alex Maucourant, François Mella, Massimiliano |
| author_facet | Calabri, Alberto Cantat, Serge Massarenti, Alex Maucourant, François Mella, Massimiliano |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic growth of the Cremona group Calabri, Alberto Cantat, Serge Massarenti, Alex Maucourant, François Mella, Massimiliano Algebraic Geometry Group Theory Primary 14E07, 14E05, Secondary 14L35, 14L40 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. |
| title | Algebraic growth of the Cremona group |
| topic | Algebraic Geometry Group Theory Primary 14E07, 14E05, Secondary 14L35, 14L40 |
| url | https://arxiv.org/abs/2503.04678 |