Curve arrangements, pencils, and Jacobian syzygies
Résumé
Let C : f = 0 be a curve arrangement in the complex projective plane. If C contains a curve subarrangement consisting of at least three members in a pencil, then we obtain an explicit syzygy among the partial derivatives of the homogeneous polynomial f. In many cases, this observation reduces the question about the freeness or the nearly freeness of C to an easy computation of Tjurina numbers. We also discuss some consequences for Terao's conjecture in the case of line arrangements and the asphericity of some complements of geometrically constructed free curves.