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 \(r\le d-1\) 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 \(r\le d-2\). This conjecture is proved for degrees \(d \le 11\).

首先，我们给出了复投影缩减平面曲线C的雅可比合合模的最小生成器集的基数m的尖锐上限。接下来，我们讨论由 A. du Plessis 和 CTC Wall 给出的上限的锐度，对于此类曲线C的全局 Tjurina 数，根据其度数d和最小度数\(r\le d-1 \)的雅可比 syzygy。我们给出了全局 Tjurina 数等于 du Plessis-Wall 上限的曲线的同调特征，这特别意味着对于这些曲线，m的上限也达到了。还给出了这些曲线在其雅可比模块的第 0 次拟合理想方面的第二个特征。最后，我们证明了某些对 ( d ,  r )具有最大全局 Tjurina 数的曲线的存在。我们推测这样的曲线对于任何对 ( d ,  r ) 都存在，此外，当\(r\le d-2\)时，它们可以被选择为线排列。这个猜想在度数\(d \le 11\) 中得到证明。