# Plane algebraic curves of arbitrary genus via Heegaard Floer homology

### Maciej Borodzik

University of Warsaw, Poland### Matthew Hedden

Michigan State University, East Lansing, USA### Charles Livingston

Indiana University, Bloomington, USA

## Abstract

Suppose $C$ is a singular curve in $CP_{2}$ and it is topologically an embedded surface of genus $g$; such curves are called cuspidal. The singularities of $C$ are cones on knots $K_{i}$. We apply Heegaard Floer theory to find new constraints on the sets of knots ${K_{i}}$ that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, $d>33$, that possess exactly one singularity which has exactly one Puiseux pair $(p;q)$. The realized triples $(p,d,q)$ are expressed as successive even terms in the Fibonacci sequence.

## Cite this article

Maciej Borodzik, Matthew Hedden, Charles Livingston, Plane algebraic curves of arbitrary genus via Heegaard Floer homology. Comment. Math. Helv. 92 (2017), no. 2, pp. 215–256

DOI 10.4171/CMH/411