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 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. ISSN 0002-9947; 1088-6850/e File URL http://www.ams.org/journals/tran/2013-365-01/S0002-9947-2012-05531-0/home.html Language English Journal Trans. Am. Math. Soc. Volume 365 Number 1 Pages 143--160 Publisher American Mathematical Society (AMS), Providence, RI Year 2013 Edition 0 Translation No Refereed No
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