  First we give a sharp upper bound for the cardinal $m$ of a minimal set of
generators for the module of Jacobian syzygies of a complex projective reduced
plane curve $C$. Next we discuss the sharpness of an upper bound, given by A.
du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$,
in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian
syzygy. We give a homological characterization of the curves whose global
Tjurina number equals the du Plessis-Wall upper bound, which implies in
particular that for such curves the upper bound for $m$ is also attained.
Finally we prove the existence of curves with maximal global Tjurina numbers
for certain pairs $(d,r)$. Moreover, we conjecture that such curves exist for
any pair $(d,r)$, and that, in addition, they may be chosen to be line
arrangements when $r\leq d-2$. This conjecture is proved for degrees $d \leq
11$.
