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# Mathematical Research Letters

## Volume 23 (2016)

### Number 2

### Lattice cohomology and rational cuspidal curves

Pages: 339 – 375

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n2.a3

#### Authors

#### Abstract

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.