  Since Schwarzenberger and his celebrated paper called "Vector bundles on the
projective plane" we know that any rank two vector bundle on $\P^2$ is a direct
image of a line bundle on a double covering of the plane. This theorem suggests
to study the rank two vector bundles according to the branch curve of the
covering which they come from. Thus, in the first part we prove that, given a
double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$,
the jumping lines of fixed order (order depending on $r$ and on the parity of
the rank two vector bundle) of the direct images vector bundles are necessarely
$r$-tangent to $C_{2r}$. In the second part we concentrate on the case $r=2$.
Then we give a list of vector bundles for which the jumping lines are exactly
the bitangent lines to the branch quartic.
