Critical cones of characteristic varieties

TL;DR

This paper explores the structure of characteristic varieties of modules over Weyl algebras, introducing a new invariant and providing bounds on their number, with implications for understanding module dimensions.

Contribution

It establishes a connection between characteristic varieties and critical cones, introduces a new invariant, and provides bounds on the number of characteristic varieties for modules over Weyl algebras.

Findings

Characteristic varieties equal to critical cones of other varieties.

All characteristic varieties from natural weights share the same dimension.

An upper bound on the number of characteristic varieties for cyclic modules.

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.