Nonisolated forms of rational triple point singularities of surfaces and their resolutions (Q309451)

Let \(S\) be a germ of a normal surface embedded in \({\mathbb C}^N\) with a singularity at the origin, and let \(\pi:\widetilde{S}\to S\) be a resolution of \(S\). The singularity of \(S\) is called rational if \(H^1(\widetilde{S},{\mathcal O}_{\widetilde{S}})=0\). This condition is known to imply a number of combinatorial results on the invariants obtained from the resolution graphs of such surfaces. In the present paper, the authors give a list of non-isolated hypersurface singularities in \({\mathbb C}^3\) such that their normalizations are rational surface singularities of multiplicity \(3\) (those can be defined by three equations in \({\mathbb C}^4\)). Using a method introduced by Oka for isolated complete intersections, they construct the corresponding minimal resolution graphs. They also show that both normal surfaces in \({\mathbb C}^4\) and their non-isolated forms in \({\mathbb C}^3\) are Newton non-degenerate, which means, roughly speaking, that they can be resolved by toric modifications well-behaved with respect to the Newton polygons.