Divisores livres em P2

Abstract

This work discusses free divisors in the projective plane. The concept of a free divisor

was introduced by K. Saito in 1980, within the context of complex analytic geometry. La- ter, an algebraic approach to the theory was developed, formulated in terms of the freeness

of the module of logarithmic derivations associated with a form f in a polynomial ring over a field. The goal of this dissertation is to present criteria that characterize when a form f defines an algebraic free divisor. Initially, classical criteria are explored, both in matrix terms, via Saito’s criterion, and from a homological perspective, using Hilbert–Burch type resolutions. As the main result of this work, we study a characterization developed by S. Tohaneanu in 2012, based on the notion of regular syzygies in the three-dimensional setting. Moreover, in the case where f defines an arrangement of hypersurfaces in P 2 , we investigate connections between the minimal degree of the syzygies of the Jacobian ideal of f and the minimal degree of the generators of the radical of this ideal.