Algebraic subgroups of the Cremona group
For a complex algebraic variety \(X\) the set \(\mathrm{Bir}(X)\) of all birational transformations of \(X\) is evidently a group. For example \(\mathrm{Bir}({\mathbb{P}}^1) \cong \mathrm{PGL}(2)\), however for \(n \geq 2\) the groups \(\mathrm{Bir}({\mathbb{P}}^n)\) are already non-algebraic, which complicates their study. A question of particular interest in the study of \(\mathrm{Bir}(X)\) is the classification of their algebraic subgroups, i.e. subgroups \(G \subset \mathrm{Bir}(X)\) for which the induced action \(G \times X \rightarrow X\) is a rational map. The classification of maximal connected algebraic subgroups \(G\) of \(\mathrm{Bir}({\mathbb{P}}^2)\) is known since the end of the 19-th century by Enriques: any such \(G\) is conjugate to the identity component of the automorphism group of a minimal rational surface. In this paper the author gives a complete classification of all maximal algebraic subgroups \(G\) of the Cremona group \(\mathrm{Bir}({\mathbb{P}}^2)\), which completes the classical results of Enriques by including the general case when \(G\) is non-connected.
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