Lattice cohomology and rational cuspidal curves

TL;DR

This paper investigates lattice cohomology related to rational cuspidal curves, providing counterexamples, verifying conjectures for specific cases, and establishing new computational tools based on lattice cohomology of surgery 3-manifolds.

Contribution

It offers a counterexample to a conjecture on rational cuspidal curves, proves a weakened conjecture for known cases, and introduces lattice cohomology methods for analyzing these curves.

Findings

Counterexample to the original conjecture with a degree 8 tricuspidal curve

Validation of a weaker conjecture for curves with up to 2 cusps

Dependence of zeroth lattice cohomology on multiplicity multisets

Abstract

We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hern\'{a}ndez and N\'{e}methi on rational cuspidal projective plane curves. The counterexample 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 and (lattice cohomology) computations of N\'emethi and Rom\'an. Then we formulate a `simplified' (slightly weaker) version, more in the spirit of the motivation of the original conjecture (comparing index type numerical invariants), and we prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomology of surgery 3-manifolds $S^3_{-d}(K)$, where $K$ is a connected sum of algebraic knots $\{K_i\}_i$. Finally, we prove that the zeroth lattice cohomology of this surgery manifold depends only on the multiset of multiplicities occurring in the multiplicity sequences describing the algebraic knots $K_i$. This result is closely related to the lattice-cohomological reformulation of the above mentioned theorems and conjectures, and provides new computational and comparison procedures.