Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces (Q330155)

The paper under review gives a description of the set of all locally nilpotent derivations of the quotient ring \(\mathcal{B} = K[X, Y, Z]/(f(X)Y-\phi(X, Z))\) constructed from the defining equation \(f(X)Y=\phi(X, Z)\) of a generalized Danielewski surface in \(\mathbb{K}^{3}\) in the case when \(\mathbb{K}\) is an algebraically closed field of characteristic zero, \(\phi(X, Z) = Z^{m} + b_{m-1}(X)Z^{m-1} +\dots + b_{1}(X)Z + b_{0}(X)\) (\(m > 1\)) and \(\deg f > 1\).NEWLINENEWLINEAs a consequence of this description, the authors obtain that both \(ML\)- and \(HD\)- invariants of \(\mathcal{B}\) introduced in [\textit{L. Makar-Limanov}, Isr. J. Math. 96, Part B, 419--429 (1996; Zbl 0896.14021)] and [\textit{H. Derksen}, Constructive invariant theory and the Linearisation problem. Ph.D. Thesis, Univ. of Basel (1997)], respectively, are equal to \(\mathbb{K}[X]\). (Recall that the Makar-Limanov (or \(ML\)-) invariant of a ring \(A\) is the intersection of the kernels of all locally nilpotent derivations of \(A\). The Derksen (or \(HD\)-) invariant of \(A\) is defined as the subring of \(A\) generated by the union of kernels of all nontrivial locally nilpotent derivations of \(A\).)NEWLINENEWLINEThe paper also presents a description of a set of generators for the group of \(\mathbb{K}\)-automorphisms of the ring \(K[X, Y, Z]/(f(X)Y-\phi(Z))\) where \(\phi(Z)\in\mathbb{K}[Z]\) and \(\deg \phi > 1\).