Characterizations of hypersurfaces of a Danielewski type (Q392481)

Let \(k\) be a field of characteristic zero. The hypersurfaces \(X_1^nX_2=\varphi(Y)\) in \(\mathbb{A}_k^3=\text{Spec}k[X_1,X_2,Y]\), where \(n\geq 1\) and \(\varphi\) is a non-constant polynomial in \(Y\), were introduced by Danielewski [``On a cancellation problem and automorphism groups of affine algebraic varieties'', Warsaw, preprint, 1989] as the counterexamples to the generalized Cancellation Problem. The coordinate ring of a Danielewski surface for which \(n=1\) admits two locally nilpotent derivations (\(D_1=\varphi'(Y)\partial_{X_2}+X_1\partial_{Y}\) and \(D_2=\varphi'(Y)\partial_{X_1}+X_2\partial_{Y}\)) with different kernels, both being polynomial rings in one variable over \(k\) (\(k[X_1]\) and \(k[X_2]\) respectively) and both satisfying \(D_i(Y)\neq 0\) and \(D_i^2(Y)=0\). In [Osaka J. Math. 41, No. 1, 37--80 (2004; Zbl 1078.13010)] \textit{D. Daigle} showed (Theorem 2.5) that the existence of such derivations characterizes these surfaces among affine \(k\)-domains and obtains another characterization among factorial (i.e.\ UFD) affine \(k\)-domains (Theorem 2.6 loc. cit.). In this article the author obtains generalizations of these criteria to the case of affine \(R\)-domains, where \(R\) is a factorial affine \(k\)-domain.