On the Hilbert vector of the Jacobian module of a plane curve (Q2211441)

Summary: We identify several classes of complex projective plane curves \(C:f=0\), for which the Hilbert vector of the Jacobian module \(N(f)\) can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on \(\mathbb{P}^2\), is used to get a sharp lower bound for the initial degree of the Jacobian module \(N(f)\), under a semistability condition.