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

2018/10/28 by Dimca, Alexandru, Sticlaru, Gabriel
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.11766

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.

Related

• Curves with Jacobian syzygies of the same degree
• Jacobian syzygies and plane curves with maximal global Tjurina numbers
• Deformations of plane curves and Jacobian syzygies
• The 0-th Fitting ideal of the Jacobian module of a plane curve
• On rational cuspidal plane curves, and the local cohomology of Jacobian rings