Recombination formulae for the spectrum of curve singularities and some applications

TL;DR

This paper derives recombination formulae for the spectra of complex plane curve singularities and applies them to generalize bounds, determine multiplicities, and analyze resolution divisors.

Contribution

It introduces new recombination formulae for spectra and applies them to generalize bounds and determine singularity invariants.

Findings

Generalized Durfee's bound

Generalized Givental's bound

Multiplicity determined by spectrum

Abstract

We obtain some recombination formulae for the spectra of (complex, reduced) plane curve singularities. As an application we prove: a generalization of Durfee's bound; a generalization of Givental's bound; the multiplicity of the curve singularity is determined by its spectrum; for many curve singularities all the multiplicities of exceptional divisors of the resolution are determined by the spectrum; etc.