The Cremona group of the plane is compactly presented (Q2793757)

It is a classical result of \textit{M. Noether} [Clebsch Ann. IV. 547--570. (1871; JFM 03.0425.02)] and \textit{G. Castelnuovo} [Torino Atti 36, 861--874 (1901; JFM 32.0675.03)] that the Cremona group \(\mathrm{Bir}(\mathbb P_{\mathbb C}^2)\) of birational selfmaps of the plane is generated by \(\mathrm{Aut}(\mathbb P^2) =\mathrm{PGL}_3(\mathbb C)\) and a single quadratic map. The relations inside this group, with respect to this system of generators or some variants, have been described successively by \textit{M. Kh. Gizatullin} [Izv. Akad. Nauk SSSR, Ser. Mat. 46, 909--970 (1982; Zbl 0509.14011)], \textit{V. A. Iskovskikh} [Russ. Math. Surv. 40, No. 5, 231--232 (1985); translation from Usp. Mat. Nauk 40, No. 5(245), 255--256 (1985; Zbl 0613.14012)] and \textit{J. Blanc} [Proc. Am. Math. Soc. 140, No. 5, 1495--1500 (2012; Zbl 1251.14008)].NEWLINENEWLINEOn the other hand, \textit{J. Blanc} and \textit{J.-P. Furter} [Ann. Math. (2) 178, No. 3, 1173--1198 (2013; Zbl 1298.14020)] extended the Euclidean topology on \(\mathrm{PGL}_3(\mathbb C)\) to a topology on the whole \(\mathrm{Bir}(\mathbb P_{\mathbb C}^2)\), such that the Cremona group becomes a Hausdorff topological group.NEWLINENEWLINEThen it is natural to ask for a compact presentation of the Cremona group, in the sense that the set of generators should be compact and the set of relations of bounded length. The paper under review precisely proposes such an explicit compact presentation.NEWLINENEWLINEThe key intermediate result, which is interesting in itself and holds true over any algebraically closed field \(K\), reads as follows:NEWLINENEWLINE Theorem. The Cremona group is generated by the three natural algebraic subgroups \(\mathrm{Aut}(\mathbb P^2)\), \(\mathrm{Aut}(\mathbb P^1 \times \mathbb P^1)\) and \(\mathrm{Aut}(\mathbb F_2)\), where \(\mathbb F_2\) is the second Hirzebruch surface. Moreover \(\mathrm{Bir}(\mathbb P_{K}^2)\) is the amalgamated product of these three subgroups along their pairwise intersections, modulo one single relation of length 4.