Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Abstract

Suppose $C$ is a singular curve in $\mathbb{C}{P}^2$ 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.