On a class of Danielewski surfaces in affine 3-space

Abstract

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 x n y = p ( z ) , Israel J. Math. 121 (2001) 113–123], L. Makar-Limanov computed the automorphism groups of surfaces in C 3 defined by the equations x n z − 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 x n z − y 2 − 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 x n z − 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 A C 3 which can be extended holomorphically but not algebraically to C ∗ -actions on A C 3 .