On Lipschitz Normally Embedded complex surface germs

TL;DR

This paper investigates Lipschitz Normally Embedded complex surface germs, establishing that their topological type determines key resolution and curve data, thus extending known results to a broader class of singularities.

Contribution

It proves that the topological type of Lipschitz Normally Embedded surface germs determines their resolution combinatorics and related curve data, generalizing previous results for minimal singularities.

Findings

Topological type determines minimal resolution combinatorics

Resolution factors through blowup of maximal ideal and Nash transform

Includes new example of Lipschitz Normally Embedded surface singularity

Abstract

We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.