Curve arrangements, pencils, and Jacobian syzygies

Let $\mathcal{C}:f=0$ be a curve arrangement in the complex projective plane. If $\mathcal{C}$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains 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 $\mathcal{C}$ to an easy computation of Tjurina numbers. Some consequences for Terao’s conjecture in the case of line arrangements are also discussed as well as the asphericity of some complements of geometrically constructed free curves. We also show that any line arrangement is a subarrangement of a free, $K(\pi ,1)$ line arrangement.