|Title||Critical cones of characteristic varieties.|
|Author(s)|| Roberto Boldini|
|Type||Article in Journal|
|Abstract||Let $ M$ be a left module over a Weyl algebra in characteristic zero. Given natural weight vectors $ \nu $ and $ \omega $, we show that the characteristic varieties arising from filtrations with weight vector $ \nu +s\omega $ stabilize to a certain variety determined by $ M$, $ \nu $, $ \omega $ as soon as the natural number $ s$ grows beyond a bound which depends only on $ M$ and $ \nu $ but not on $ \omega $.|
As a consequence, in the notable case when $ \nu $ is the standard weight vector, these characteristic varieties deform to the critical cone of the $ \omega $-characteristic variety of $ M$ as soon as $ s$ grows beyond an invariant of $ M$.
As an application, we give a new, easy, non-homological proof of a classical result, namely, that the $ \omega $-characteristic varieties of $ M$ all have the same Krull dimension.
The set of all $ \omega $-characteristic varieties of $ M$ is finite. We provide an upper bound for its cardinality in terms of supports of universal Gröbner bases in the case when $ M$ is cyclic. By the above stability result, we conjecture a second upper bound in terms of total degrees of universal Gröbner bases and of Fibonacci numbers in the case when $ M$ is cyclic over the first Weyl algebra.
|Journal||Trans. Am. Math. Soc.|
|Publisher||American Mathematical Society (AMS), Providence, RI|