  For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$,
we study the set of jumping lines for the rank two vector bundle $T\langle C
\rangle $ on $\mathbb{P}^2$, whose sections are the logarithmic vector fields
along $C$. We point out the relations of these jumping lines with the Lefschetz
type properties of the Jacobian module of $f$ and with the Bourbaki ideal of
the module of Jacobian syzygies of $f$. In particular, when the vector bundle
$T\langle C \rangle $ is unstable, a line is a jumping line if and only if it
meets the 0-dimensional subscheme defined by this Bourbaki ideal, a result
going back to Schwarzenberger. Other classical general results by Barth,
Hartshorne and Hulek resurface in the study of this special class of rank two
vector bundles.
