Finite abelian subgroups of the Cremona group of the plane

Details Finite abelian subgroups of the Cremona group of the plane Finite abelian subgroups of the Cremona group of the plane

2006-10-11

arxiv.org/abs/math/0610368v1

Mathematics

Algebraic Geometry

PHD Thesis, 189 pages, 34 figures, original text may be found at http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.htm

Scientific paper

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on birational transformations, as for example the existence of infinitely many conjugacy classes of elements of order n, for any even number n, a result false in the odd case. We prove also that a root of some linear transformation of finite order is itself conjugate to a linear transformation.

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