Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Abstract

Suppose C is a singular curve in \mathbb 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.

Locations

• Commentarii Mathematici Helvetici - View
• arXiv (Cornell University) - View - PDF
• IUScholarWorks Open (Indiana University) - View - PDF