Volume 25, issue 7 (2021)

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Factorization statistics and bug-eyed configuration spaces

Dan Petersen and Philip Tosteson

Geometry & Topology 25 (2021) 3691–3723
Abstract

A recent theorem of Hyde proves that the factorization 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 ${ℝ}^{3}$. 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 any Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

Keywords
arithmetic topology, configuration spaces, hyperplane arrangement, Salvetti complex
Mathematical Subject Classification
Primary: 11T06, 14F20, 14N20, 55R80
Secondary: 14A20, 14G15