For line arrangements, the minimal degree of a logarithmic derivation changes predictably under adding or deleting one line, yielding new maximal Tjurina arrangements and a sharp n3≤5 combinatoriality threshold.
Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves
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abstract
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.
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Addition-deletion results for the minimal degree of logarithmic derivations of arrangements
For line arrangements, the minimal degree of a logarithmic derivation changes predictably under adding or deleting one line, yielding new maximal Tjurina arrangements and a sharp n3≤5 combinatoriality threshold.