Curve arrangements, pencils, and Jacobian syzygies

TL;DR

This paper explores the relationship between curve arrangements, pencils, and Jacobian syzygies, providing explicit methods to analyze freeness and nearly freeness of arrangements, with implications for Terao's conjecture and topological properties.

Contribution

It introduces a method to derive explicit syzygies from pencil subarrangements, simplifying the study of freeness and nearly freeness of curve arrangements.

Findings

Explicit syzygies from pencil subarrangements

Reduction of freeness questions to Tjurina number computations

Any line arrangement is contained in a free, aspherical line arrangement

Abstract

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.