Automorphisms of open surfaces with irreducible boundary

Abstract

Let $(S, B_{S})$ be the log pair associated with a projective completion of a smooth quasi-projective surface $V$. Under the assumption that the boundary $B_{S}$ is irreducible, we obtain an algorithm to factorize any automorphism of $V$ into a sequence of simple 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 Jonquières 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 Jonquières automorphisms, which we call the tame group of $V$, is a proper subgroup of $\mathrm{Aut}(V)$.