On a class of Danielewski surfaces in affine 3-space

In [L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990) 250–256] and [L. Makar-Limanov, On the group of automorphisms of a surface xny=p(z), Israel J. Math. 121 (2001) 113–123], L. Makar-Limanov computed the automorphism groups of surfaces in C3 defined by the equations xnz−P(y)=0, where n⩾1 and P(y) is a nonzero polynomial. Similar results have been obtained by A. Crachiola [A. Crachiola, On automorphisms of Danielewski surfaces, J. Algebraic Geom. 15 (2006) 111–132] for surfaces with equations xnz−y2−σ(x)y=0, where n⩾2 and σ(0)≠0, defined over arbitrary base fields. Here we consider more general surfaces defined by equations xnz−Q(x,y)=0, where n⩾2 and Q(x,y) is a polynomial with coefficients in an arbitrary base field k. We characterize among them the ones which are Danielewski surfaces in the sense of [A. Dubouloz, Danielewski–Fieseler surfaces, Transformation Groups 10 (2) (2005) 139–162], and we compute their automorphism groups. We study closed embeddings of these surfaces in affine 3-space. We show that in general their automorphisms do not extend to automorphisms of the ambient space. Finally, we give explicit examples of C∗-actions on a surface in which can be extended holomorphically but not algebraically to C∗-actions on .