2007/06/15 by Hubert Flenner, Shulim Kaliman, Flenner, Hubert +3
Mathematics · #14R20 #14R25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.AG #msc:14R20 #msc:14R25
paper · pdf · doi:10.48550/arxiv.0706.2261
43 pages
arxiv created 2007/06/15 · openalex publication_date 2007/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A Gizatullin surface is a normal affine surface V over \bf C, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of \bf C^*-actions and \bf A1-fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with \bf C+-actions on V considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one \bf A1-fibration V→ S up to an isomorphism of the base S. Moreover an effective \bf C*-action on them, if it does exist, is unique up to conjugation and inversion t↦ t-1 of \bf C^*. Obviously uniqueness of \bf C^*-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of \bf A1-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of \bf C^*-actions and \bf A1-fibrations. In the present paper we obtain a criterion as to when \bf A1-fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base S. We exhibit as well a large subclasses of Gizatullin \bf C*-surfaces for which a \bf C^*-action is essentially unique and for which there are at most two conjugacy classes of \bf A1-fibrations over \bf A1.