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Linearisation of finite Abelian subgroups of the Cremona group of the plane

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Published in Groups Geometry and Dynamics. 2009, vol. 3, no. 2, p. 215-232
Abstract Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it 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 none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to Z/2Z×Z/4Z, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.
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BLANC, Jeremy. Linearisation of finite Abelian subgroups of the Cremona group of the plane. In: Groups Geometry and Dynamics, 2009, vol. 3, n° 2, p. 215-232. https://archive-ouverte.unige.ch/unige:8510

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Deposited on : 2010-07-01

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