JournalscmhVol. 92 , No. 2pp. 215–256

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

  • Maciej Borodzik

    University of Warsaw, Poland
  • Matthew Hedden

    Michigan State University, East Lansing, USA
  • Charles Livingston

    Indiana University, Bloomington, USA
Plane algebraic curves of arbitrary genus via Heegaard Floer homology cover
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Abstract

Suppose CC is a singular curve in CP2\mathbb CP^2 and it is topologically an embedded surface of genus gg; such curves are called cuspidal. The singularities of CC are cones on knots KiK_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {Ki}\{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>33d > 33, that possess exactly one singularity which has exactly one Puiseux pair (p;q)(p;q). The realized triples (p,d,q)(p,d,q) are expressed as successive even terms in the Fibonacci sequence.

Cite this article

Maciej Borodzik, Matthew Hedden, Charles Livingston, Plane algebraic curves of arbitrary genus via Heegaard Floer homology. Comment. Math. Helv. 92 (2017), no. 2 pp. 215–256

DOI 10.4171/CMH/411