## 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:

 1.4e+08 Birational automorphisms, Cremona group and generalizations 1.4e+21 Coverings in algebraic geometry

### Citations:

Zbl 1055.14012; Zbl 1057.14053
Full Text:

### References:

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