Uniqueness of ${\mathbf{C}}^{\ast }$- and ${\mathbf{C}}_{+}$-actions on Gizatullin surfaces

Abstract

A Gizatullin surface is a normal affine surface $V$ over $\mathbf{C}$, 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 ${\mathbf{C}}^{\ast }$-actions and ${\mathbf{A}}^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with ${\mathbf{C}}_{+}$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one ${\mathbf{A}}^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective ${\mathbf{C}}^{\ast }$-action on them, if it does exist, is unique up to conjugation and inversion $t\mapsto t^{-1}$ of ${\mathbf{C}}^{\ast }$. Obviously uniqueness of ${\mathbf{C}}^{\ast }$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of ${\mathbf{A}}^1$-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 ${\mathbf{C}}^{\ast }$-actions and ${\mathbf{A}}^1$-fibrations. In the present paper we obtain a criterion as to when ${\mathbf{A}}^1$-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 ${\mathbf{C}}^{\ast }$-surfaces for which a ${\mathbf{C}}^{\ast }$-action is essentially unique and for which there are at most two conjugacy classes of ${\mathbf{A}}^1$-fibrations over ${\mathbf{A}}^1$.

43 pages

References in corpus (2)

• Locally nilpotent derivations on affine surfaces with a $\C^*$-action
• On the uniqueness of ${\mathbf{C}}^{\ast }$-actions on affine surfaces