Critical cones of characteristic varieties

Abstract

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.