On a class of Danielewski surfaces in affine 3-space
Résumé
L. Makar-Limanov computed the automorphisms 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 for surfaces defined by the equations xnz−y2−h(x)y=0, where n≥2 and h(0)≠0, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations xnz−Q(x,y)=0, where n≥2 and Q(x,y) is a polynomial with coefficients in an arbitrary base field k. Among these surfaces, we characterize the ones which are Danielewski surfaces 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 the ambient space. Finally, we give explicit examples of C∗-actions on a surface in C3 which can be extended holomorphically but not algebraically to a C∗-action on C3.