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Commentarii Mathematici Helvetici

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Volume 92, Issue 2, 2017, pp. 215–256
DOI: 10.4171/CMH/411

Published online: 2017-05-22

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Maciej Borodzik[1], Matthew Hedden[2] and Charles Livingston[3]

(1) University of Warsaw, Poland
(2) Michigan State University, East Lansing, USA
(3) Indiana University, Bloomington, USA

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.

Keywords: Complex plane curves, $d$-invariants, Heegaard Floer homology

Borodzik Maciej, Hedden Matthew, Livingston Charles: Plane algebraic curves of arbitrary genus via Heegaard Floer homology. Comment. Math. Helv. 92 (2017), 215-256. doi: 10.4171/CMH/411