On the Cohomology with local coefficients of Pure Braid Groups

The need to calculate local system cohomolgy of the complement of an hyperplane arrangement arises in various contexts. Nevertheless until now very few it’s known on the direct computation of such cohomology. In this paper the author describes all generators of the first homology group of braid arrangement giving also a complete description of its first characteristic variety, fulfilling results obtained by D. Cohen and A. Suciu in [7]. Moreover she gives also a complete description of the first characteristic variety in the case of the generalized braid arrangement coming from the dihedral group.