  We show that certain characteristic varieties of a finitely generated module
over a given Weyl algebra arising from weighted degree filtrations are equal to
the critical cone of some other characteristic varieties. This behaviour of the
characteristic varieties permits us to introduce a new invariant of the module.
As a second consequence we are able to provide an easy and non-homological
proof that the characteristic varieties of a module arising from weights in the
natural polynomial region of the Weyl algebra all have the same Krull and
Gelfand-Kirillov dimension, equal to the Gelfand-Kirillov dimension of the
module. Third we give an upper bound for the number of distinct characteristic
varieties of a cyclic module in terms of degrees of elements in universal
Groebner bases and the above results allow us to conjecture a further upper
bound.
