Cuspidal curves and Heegaard Floer homology (Q2800462)

Let \(C\) be an unicuspidal curve of degree \(d\) and genus \(g\). The main result in the paper is to prove that NEWLINE\[NEWLINE k-g \leq I_{jd+1-2k}- \frac{(d-j-2)(d-j-1)}{2} \leq k, NEWLINE\]NEWLINE for every \(-1 \leq j \leq d-2\) and every \(0 \leq k \leq g\), where \(I_m\) is the number of gaps of the semigroup of the singularity in \([m, + \infty)\). This result has been independently proved by \textit{M. Borodzik}, \textit{M. Hedden} and \textit{C. Livingston} [``Plane algebraic curves of arbitrary genus via Heegaard Floer homology'', Preprint, \\url{arXiv:1409.2111}] and extends another by \textit{M. Borodzik} and \textit{C. Livingston} [Forum Math. Sigma 2, Article ID e28, 23 p. (2014; Zbl 1325.14047)] which proves a conjecture by Bobadilla, Luengo, Melle-Hernández and Nèmethi [\textit{J. Fernández De Bobadilla} et al., Proc. Lond. Math. Soc. (3) 92, No. 1, 99--138 (2006; Zbl 1115.14021)]. A generalization for curves with more cusps is also given. As an interesting application, the authors prove that for \(C\) unicuspidal with \(g \leq 1\) and only a Puiseux pair, with singularity of type \((a,b)\), it holds that \(a+b=3d\) if \(d\) is large enough. They also show that, as a consequence, for all genera \(g\) such that \(g \cong 2 \pmod{5}\) or \(g \cong 4 \pmod{5}\), there are only finitely many unicuspidal curves of genus \(g\) up to equisingularity.