Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces

A Pfaff field on P(k)(n) is a map eta : Omega(s)(Pkn) -> L from the sheaf of differential s-forms to an invertible sheaf. The interesting ones are those arising from a Pfaff system, as they give rise to a distribution away from their singular locus. A subscheme X subset of P(k)(n) is said to be invariant under if eta induces a Pfaff field Omega(s)(X) -> L vertical bar x. We give bounds for the Castelnuovo-Mumford regularity of invariant complete intersection subschemes (more generally, arithmetically Cohen-Macaulay subschemes) of dimension s, depending on how singular these schemes are, thus bounding the degrees of the hypersurfaces that cut them out.