Elements and cyclic subgroups of finite order of the Cremona group. (English) Zbl 1213.14029

The Cremona group \(\mathrm{Bir}(\mathbb{P}^2)\) is the group of birational transformations of 2. An element of \(\mathrm{Bir}(\mathbb{P}^2)\) is said to be of de Jonquières if it preserves a pencil of rational curves. The classification of finite cyclic subgroups of prime order of \(\mathrm{Bir}(\mathbb{P}^2)\) was totally achieved a few years ago, see e.g. [A. Beauville and J. Blanc, C. R., Math., Acad. Sci. Paris 339, No. 4, 257–259 (2004; Zbl 1062.14017)]. The classification of all finite cyclic subgroups which are not of de Jonquières was almost achieved in [I. V. Dolgachev and V. A. Isovskikh, Progress in Mathematics 269, 443–548 (2009; Zbl 1219.14015)], where a list of representative elements is available (ibid. Table 9); explicit forms are given and the dimension of the varieties which parametrise the conjugacy classes are also provided; there are 29 families of groups of order at most \(30\). In the present article is achieved the classification of elements and cyclic subgroups of finite order of \(\mathrm{Bir}(\mathbb{P}^2)\). The main work concerns de Jonquières elements, see Theorem 2. Next, Theorem 3 refines the results of Dolgachev and Iskovskikh, by providing the parametrisations of the \(29\) families of cyclic groups which are not of de Jonquières type.


14E07 Birational automorphisms, Cremona group and generalizations
20E45 Conjugacy classes for groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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