On the module of derivations of a line arrangement

To each multiple point $p$ in a line arrangement $\mathcal A$ in the complex projective plane we associate a local derivation $\tilde D_p \in D_0( \mathcal A)$. We show first that these derivations span the graded module of derivations $D_0( \mathcal A)$ in all degrees $\geq d -3$, where $d$ is the number of lines in $\mathcal A$, see Theorem 1.4 and Theorem 1.6. Then, to each local derivation $\tilde D_p \in D_0( \mathcal A)$ we associate a polynomial $g_p$ which seems to play a key role in the characterization of the freeness of $\mathcal A$, see Theorem 1.10, as well as in the study of the position of the multiple points of $\mathcal A$ with respect to unions of lines, see Corollary 1.13 and Conjecture 1.14. Corollary 1.9 gives a result of an independent interest, namely a lower bound for the maximal exponent of a plane curve having a line as an irreducible component.