On Cremona transformations of prime order.(English)Zbl 1062.14017

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 T. de Fernex [Nagoya Math. J. 174, 1–28 (2004; Zbl 1062.14019)] (more general surfaces). The article under review answers a question in de Fernex’s paper (page 8, theorem E) by showing that the order-5 element in [ibid., example E4] is conjugate to a linear automorphism of $${\mathbb P}^2$$. Thus it is concluded that every non-linear element of order 5 in Bir$$({\mathbb P}^2)$$ is conjugate to a unique element given in [ibid, example E3]. As a consequence of the de Fernex’s classification and the result of the current article, it is true that a birational transformation of prime order is not conjugate to a linear automorphism if and only if it fixes an irrational curve (not true for non-prime order case).

MSC:

 14E07 Birational automorphisms, Cremona group and generalizations 14J26 Rational and ruled surfaces 14J50 Automorphisms of surfaces and higher-dimensional varieties
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References:

 [1] Bayle, L.; Beauville, A., Birational involutions of $$\mathbb{P}^2$$. Kodaira’s issue, Asian J. math., 4, 11-17, (2000) · Zbl 1055.14012 [2] de Fernex, T., On planar Cremona maps of prime order, Nagoya math. J., 174, (2004) · Zbl 1062.14019 [3] Semple, J.; Roth, L., Introduction to algebraic geometry, (1949), Clarendon Oxford · Zbl 0041.27903
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