A hyperplane arrangement A is a finite union of hyperplanes in projective space. Thus, it is defined by a product f of linear forms. Its Jacobian ideal J is generated by the partial derivatives of f, and A is said to be free if J is Cohen-Macaulay. Terao's conjecture posits that freeness is a combinatorial property of a hyperplane arrangements, that is, it is determined by the intersection lattice of A. Extending earlier results, we discuss results in the spirit of this conjecture. In particular, we identify mild conditions on A, which imply that the graded Betti numbers of the top-dimensional part of J are combinatorially determined.

For line arrangements, we have additional results, including a new freeness criterion.

The talk is based on ongoing joint work with Juan Migliore.