Nonisolated forms of rational triple point singularities of surfaces and their resolutions

TL;DR

This paper investigates nonisolated rational triple point surface singularities, providing a classification, constructing their minimal resolutions via Newton polygon subdivisions, and demonstrating their Newton non-degeneracy.

Contribution

It introduces a classification of nonisolated hypersurface singularities related to RTP-singularities and develops a method to construct their minimal resolutions using Newton polygons.

Findings

List of nonisolated hypersurface singularities with RTP-normalisations

Construction of minimal resolution graphs via Newton polygons

Proof of Newton non-degeneracy for these singularities

Abstract

The work is a detailed study of rational singularities of multiplicity 3 (RTP-singularities, for short). We give a list of nonisolated hypersurface singularities of which normalisations are the RTP-singularities, and construct their minimal resolution graphs by means of a subdivision of Newton polygons of those -- a method introduced by M. Oka for isolated complete intersection singularities. We show that nonisolated forms of RTP-singularities and their normalisations are both Newton non-degenerate.