Abstract
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.
Citation
Julie Déserti. "On solvable subgroups of the Cremona group." Illinois J. Math. 59 (2) 345 - 358, Summer 2015. https://doi.org/10.1215/ijm/1462450705
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