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

## TL;DR

This paper characterizes the Hilbert vector of the Jacobian module for specific classes of plane curves, providing explicit formulas and bounds using vector bundle cohomology techniques.

## Contribution

It identifies classes of curves with fully determined Jacobian module Hilbert vectors and establishes a sharp lower bound for the initial degree under semistability.

## Key findings

- Complete determination of Hilbert vectors for 3-syzygy, maximal Tjurina, and nodal rational curves.
- A sharp lower bound for the initial degree of the Jacobian module under semistability.
- Application of Hartshorne's cohomology results to curve Jacobian modules.

## Abstract

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.