  Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal
degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal
degree of a derivation killing $f$, is an important invariant of the curve $C$,
for instance it can be used to determined whether $C$ is free or nearly free.
In this note we study the relations of this invariant $mdr(f)$ with a
decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common
irreducible components. When all the singularities that occur are
quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer
information on this invariant in this setting. Using this, we give some
geometrical criteria, the first ones of this type in the existing literature as
far as we know, for a line to be a jumping line for the rank 2 vector bundle of
logarithmic vector fields along a reduced curve $C$.
