Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Maciej Borodzik; Matthew Hedden; Charles Livingston

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arXiv · 1409.2111

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Maciej Borodzik·Matthew Hedden·Charles Livingston

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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.

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Maciej Borodzik, Matthew Hedden, Charles Livingston. 2014-09-07. Plane algebraic curves of arbitrary genus via Heegaard Floer homology. https://arxiv.org/abs/1409.2111

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