Graded derivation modules and algebraic free divisors

The main purpose of this paper is to furnish new criteria for freeness of (algebraic, homogeneous) divisors, especially by means of the minimal number of generators of certain graded derivation modules. Our approach is based on the description of the graded syzygies of the derivation module in the hypersurface case, which allows us to derive several other applications. We investigate, under certain conditions, the Castelnuovo–Mumford regularity and the Hilbert function of such module, as well as an Eisenbud matrix factorization of the given polynomial. We also obtain the defining ideals of the blowup algebras of the derivation module, as a dual version, in the hypersurface case, of the so-called tangent algebras  introduced by Simis, Ulrich and Vasconcelos. Finally, we give an explicit Ulrich ideal and the Hilbert polynomial in the distinguished case of linear free divisors (in the sense of Buchweitz and Mond).