On the Zariski-Lipman conjecture for normal algebraic surfaces (Q2874659)

Given an algebraic variety \(V\) over a field \(k\) of characteristic zero, the following are two equivalent formulations of the Zariski-Lipman Conjecture.NEWLINENEWLINE\textsl{(1) Let \(p\) be a (closed) point of \(V\) with local ring \(R\). If the module of \(k\)-derivations \(\text{Der}_k(R)\) is a free \(R\)-module, then \(V\) is smooth at \(p\).}NEWLINENEWLINE(2) \textsl{If \(V\) is normal and the tangent bundle of the smooth locus of \(V\) is a trivial bundle, then \(V\) is smooth.}NEWLINENEWLINEThe authors prove instances of (the second formulation of) the conjecture in dimension two, using the theory of non-complete algebraic surfaces by the Japanese school. More precisely, denote by \(V^0\) the smooth locus of an algebraic surface \(V\) and by \(\bar \kappa (V^0)\) its logarithmic Kodaira dimension. Assuming that moreover \(k\) is algebraically closed and that the tangent bundle of \(V^0\) is trivial, the authors show the following. {\parindent=6mm \begin{itemize} \item[(1)] If \(V\) is affine and \(\bar \kappa (V^0) \leq 1\), then \(V\) is smooth. \item [(2)] If \(V\) is projective, then \(\bar \kappa (V^0) \leq 0\) and \(V\) has at most one singularity. \item [(3)] If \(V\) is projective and \(\bar \kappa (V^0) =0\), then \(V\) is smooth. NEWLINENEWLINE\end{itemize}} Moreover, if \(V\) is projective and \(\bar \kappa (V^0) =-\infty\), they construct a \(\mathbb P^1\)- fibration \(W \to C\), where \(W\) is a resolution of singularities of \(V\) and \(C\) is a smooth projective curve. In this setting they `almost always\'\ prove that \(V\) is smooth, that is, except when this fibration has a certain very special form.