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

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.

v8: a small change in title; Theorem 3.5 is new, and gives a characterization of curves whose global Tjurina numbers equal the lower bound obtained by A. du Pleases and C.T.C Wall

References in corpus (3)

• On the splitting types of bundles of logarithmic vector fields along plane curves
• Plus-one generated and next to free arrangements of hyperplanes
• On the cohomology of rank two vector bundles on P2 and a theorem of Chiantini and Valabrega

Cited by in corpus (2)

• Quasi-complete intersections in P2 and syzygies
• On some freeness-type properties for line arrangements