  A recent theorem of Hyde proves that the factorizations statistics of a
random polynomial over a finite field are governed by the action of the
symmetric group on the configuration space of $n$ distinct ordered points in
$\mathbb R^3$. Hyde asked whether this result could be explained geometrically.
We give a geometric proof of Hyde's theorem as an instance of the
Grothendieck--Lefschetz trace formula applied to an interesting, highly
nonseparated algebraic space. An advantage of our method is that it generalizes
uniformly to an arbitrary Weyl group. In the process we study certain
non-Hausdorff models for complements of hyperplane arrangements, first
introduced by Proudfoot.
