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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.

MSC:

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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