Linearisation of Finite Abelian Subgroups of the Cremona Group of the Plane
نویسنده
چکیده
This article gives the proof of results announced in [Bla07a] and some description of automorphisms of rational surfaces. Given a finite abelian subgroup of the Cremona group of the plane, we give a way to decide whether this one is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if neither of its elements fixes a non-rational curve. For finite abelian groups, there exists only one amazing counterexample, which is a group isomorphic to Z/2Z × Z/4Z, whose nontrivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. Some description of automorphisms (not necessarily of finite order) of Del Pezzo surfaces and conic bundles are also given.
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