F-jumping and F-Jacobian ideals for hypersurfaces

We introduce two families of ideals, F-jumping ideals and F-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the D-modules Mα that were introduced by Blickle, Mustaţă and Smith. Using strong connections between F-jumping ideals and generalized test ideals, we give a characterization of F-jumping numbers for hypersurfaces via D-modules and F-modules. In addition, we use F-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for F-regularity. Moreover, we prove several properties of F-Jacobian ideals that resemble those of Jacobian ideals of polynomials.