Cuspidal curves and Heegaard Floer homology

TL;DR

This paper establishes bounds on singularity gap functions of cuspidal plane curves, explores their implications for unicuspidal curves with one Puiseux pair, and provides classifications and examples.

Contribution

It generalizes bounds on singularity gaps, derives identities linking singularity parameters with curve degree and genus, and advances classification of cuspidal curves.

Findings

Derived bounds on gap functions for cuspidal singularities.

Proved identities relating singularity parameters, genus, and degree.

Constructed infinite families and provided classification results.

Abstract

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.