On the projective dimension of some deformations of Weyl arrangements

Abstract

We show that the logarithmic derivation module of (the cone of) the deformation A of a Weyl arrangement associated with a root system of simply laced type has projective dimension one if the deforming parameter ranges from -j to j+2. In addition, we give an explicit minimal free resolution when the root system is of type A3 and B2. Moreover, in the second case, we determine the jumping lines of maximal jumping order of the associated vector bundle. When the deforming parameter of A (respectively A') ranges from -k to k+j (respectively, from -k' to k'+j), with k different from k' and j at least 3, this allows to distinguish D0(A) from D0(A') shifted by 4(k'-k), even though these modules have the same graded Betti numbers.