Plane algebraic curves of arbitrary genus via Heegaard Floer homology

TL;DR

This paper uses Heegaard Floer homology to establish new constraints on the types of knots that can appear as singularities of algebraic curves in the complex projective plane, leading to solutions for specific genus and degree cases.

Contribution

It introduces a novel application of Heegaard Floer theory to algebraic geometry, providing constraints on singularity links and solving existence problems for certain cuspidal curves.

Findings

Constraints on knots as singularity links derived from Floer homology

Classification of genus one curves with degree greater than 33 and one singularity

Identification of realized (p,d,q) triples as Fibonacci sequence terms

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.