Resumen de Jacobian syzygies, Fitting ideals, and plane curves with maximal global Tjurina numbers
Alexandru Dimca, Gabriel Sticlaru
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≤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. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). 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≤d−2. This conjecture is proved for degrees d≤11.
