Lattice cohomology and rational cuspidal curves

We show a counterexample to a conjecture of Fernández de Bobadilla, Luengo, Melle-Hernández and Némethi on rational cuspidal projective plane curves, formulated in [9]. The counter-example is a tricuspidal curve of degree $8$. On the other hand, we show that if the number of cusps is at most $2$, then the original conjecture can be deduced from the recent results of Borodzik and Livingston [3] and the computations of [24] by the second author and Román.

We also formulate a weaker conjecture and prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomologies of certain surgery $3$-manifolds.

Finally, we study the behaviour of the semigroup counting function of an irreducible plane curve singularity under blowing up in terms of its multiplicity sequence. As a corollary, we obtain a stability result of the $0$th lattice cohomology of certain surgery $3$-manifolds with respect to certain manipulation of the multiplicity sequences of the knots.

Keywords

rational cuspidal curves, superisolated singularities, lattice cohomology, normal surface singularities, hypersurface singularities, links of singularities, geometric genus, plumbing graphs, $\mathbb{Q}$-homology spheres, Seiberg–Witten invariant

2010 Mathematics Subject Classification

Primary 32S05, 32S25, 32S50, 57M27. Secondary 14Bxx, 32Sxx, 55N35, 57R57.

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Accepted 27 January 2016

Published 6 June 2016