On Isolated Rational Singularities of Surfaces

In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These configurations arise naturally in other contexts, for instance as exceptional curves for pluricanonical embeddings of surfaces [7], and so it seems desirable to have a converse result, showing that a singularity giving rise to such a configuration is necessarily a double point. We have reconsidered the question in a more general context, and obtain in addition the correct nunferical characterization of singularities (cf. definition below). This characterization is made without the assumption that the surface is embedded and points out the connection of Du Val's work with Castelnuovo's criterion for exceptional curves [2], ([8], p. 38). Finally, we list the configurations obtained from rational triple points.