On the Hilbert vector of the Jacobian module of a plane curve

We identify several classes of complex projective plane curves C : f ¼ 0, for which the Hilbert vector of the Jacobian module NðfÞ 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 P2 , is used to get a sharp lower bound for the initial degree of the Jacobian module NðfÞ, under a semistability condition.