On a certain generalization of triangle singularities

TL;DR

This paper generalizes triangle singularities to higher dimensions, proves finiteness of hypersurface cases in each dimension, and provides a complete classification in dimension three.

Contribution

It introduces a higher-dimensional generalization of triangle singularities and classifies all hypersurface instances in three dimensions.

Findings

Finitely many hypersurface singularities in each dimension

Complete classification of these singularities in dimension three

Extension of classical triangle singularities to higher dimensions

Abstract

Triangle singularities are Fuchsian singularities associated with von Dyck groups, which are index two subgroups of Schwarz triangle groups. Hypersurface triangle singularities are classified by Dolgachev, and give 14 exceptional unimodal singularities classified by Arnold. We introduce a generalization of triangle singularities to higher dimensions, show that there are only finitely many hypersurface singularities of this type in each dimension, and give a complete list in dimension 3.