Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.