  We identify several classes of 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.
