Free Arrangements and de Rham Cohomology of Milnor Fibers

Abstract

We consider free projective hypersurfaces, stressing the relation with the Jacobian syzygies of the defining equation and giving a proof of K. Saito’s Criterion in this setting. The factorization property for $\pi ({\mathscr {A}}, t)$ when ${\mathscr {A}}$ is a free arrangement, the fact that any supersolvable arrangement is free, and the freeness of reflection arrangements are all stated in the first section. Then we restrict to the case of curves (and in particular, line arrangements) in ${\mathbb {P}}^2$, and state several characterizations of such free curves. Some applications to H. Terao’s conjecture are also given. Next we introduce a spectral sequence approach to the computation of the Alexander polynomial of a plane curve. Two algorithms are described, one for free plane curves, the other for curves in ${\mathbb {P}}^2$ having only weighted homogeneous singularities.