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On planar Cremona maps of prime order. (English) Zbl 1062.14019

The Cremona group \(\text{Bir}(\mathbb{P}^2)\) has been studied for a long time. Their modern treatment, using Mori theory, has been done in L. Bayle and A. Beauville [Asian J Math 4, 11–17 (2000; Zbl 1055.14012)] (order \(n = 2\)), D.-Q. Zhang [J. Algebra 238, 560–589 (2001; Zbl 1057.14053)] (general \(n\)) and the current paper under review. The result in theorems A and B of the article is the classification of minimal pairs \((X, \sigma)\) where \(X\) is a smooth projective surface with non-nef \(K_X\) and \(\sigma \in \text{Aut}(X)\) is of prime order \(p\); see the article for the list (the equations of the surfaces and the actions on the coordinates are also given). As a consequence, the author gives the complete classification of Cremona transformations of prime order in \(\mathbb{P}^2\) (theorem E), where the un-resolved case \(E4\) has been proved by Bayle-Beauville [loc. cit.] to be conjugate to a linear automorphism of \(\mathbb{P}^2\). The author also describes (theorem F) the moduli spaces of conjugacy classes of prime order cyclic subgroups of \(\text{Bir}(\mathbb{P}^2)\). The reader will gain a lot more by reading the article.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14E20 Coverings in algebraic geometry
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