Heegaard Floer homology and rational cuspidal curves

TL;DR

This paper uses Heegaard Floer homology to analyze the topology of complex curves in the projective plane, resolving a conjecture about Alexander polynomials for genus 0 curves with a single singular point.

Contribution

It introduces a novel application of Heegaard Floer homology to determine topological invariants of complex algebraic curves, resolving an open conjecture.

Findings

Resolved the conjecture for genus 0 curves with one singular point

Established topological constraints on Alexander polynomials of singular links

Extended results to curves with multiple singular points

Abstract

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus 0. Generalizations apply in the case of multiple singular points.