Jacobian syzygies, fitting ideals, and plane curves with maximal global Tjurina numbers (Q2164947)

This work makes a meticulous investigation about special plane curves, those named by the authors as maximal Tjurina curves of type \((d,r)\). In this context, important characterizations were made about the existence of such curves, especially those in which \(2r \geq d\), as well as studying it from a homological point of view. In particular, it was shown that if \(m = 2r - d + 3\) then a maximal Tjurina curve of type \((d,r)\) is exactly an \(m\)-syzygy curve. In addition, a description of the \(0\)-th Fitting ideal \(\mathrm{Fitt}_0(N(f))\) of the Jacobian module \(N(f)\) of any reduced plane curve \(C: f = 0 \) was performed in terms of the first and second order syzygies of the Jacobian ideal \(J_f\). The authors conjectured the existence of curves with maximal global Tjurina numbers, and chosen to be line arrangements when \(r \leq d-2\), and they demonstrated this conjecture in the case that \(d \leq 11\).