Automorphisms of open surfaces with irreducible boundary

Adrien Dubouloz (1) , Stéphane Lamy (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

Let (S, B) be the log pair associated with a projective completion of a smooth quasi-projective surface V . Under the assumption that the boundary B is irreducible, we obtain an algorithm to factorize any automorphism of V into a sequence of simple birational links. This factorization lies in the framework of the log Mori theory, with the property that all the blow-ups and contractions involved in the process occur on the boundary. When the completion S is smooth, we obtain a description of the automorphisms of V which is reminiscent of a presentation by generators and relations except that the "generators" are no longer automorphisms. They are instead isomorphisms between different models of V preserving certain rational fibrations. This description enables one to define normal forms of automorphisms and leads in particular to a natural generalization of the usual notions of affine and Jonquieres automorphisms of the affine plane. When V is affine, we show however that except for a finite family of surfaces including the affine plane, the group generated by these affine and Jonquieres automorphisms, which we call the tame group of V , is a proper subgroup of Aut(V ).

Mots clés: open surfaces; automorphisms; normal forms; log-MMP

Domaines: Géométrie algébrique [math.AG]

Adrien Dubouloz, Stéphane Lamy. Automorphisms of open surfaces with irreducible boundary. Osaka Journal of Mathematics, 2015, 52 (3), pp.747-791. ⟨hal-00980783⟩