The Bourbaki Degree of Line Arrangements

Source abstract

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition--deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.