Illinois Journal of Mathematics

On solvable subgroups of the Cremona group

Julie Déserti

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The Cremona group $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is the group of birational self-maps of $\mathbb{P}^{2}_{\mathbb{C}}$. Using the action of $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ on the Picard-Manin space of $\mathbb{P}^{2}_{\mathbb{C}}$, we characterize its solvable subgroups. If $\mathrm{G}\subset\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is solvable, nonvirtually Abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $\mathrm{G}$ is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.

We also give some corollaries.

Article information

Illinois J. Math., Volume 59, Number 2 (2015), 345-358.

Received: 9 April 2015
Revised: 3 February 2016
First available in Project Euclid: 5 May 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E07: Birational automorphisms, Cremona group and generalizations 14E05: Rational and birational maps


Déserti, Julie. On solvable subgroups of the Cremona group. Illinois J. Math. 59 (2015), no. 2, 345--358. doi:10.1215/ijm/1462450705.

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