Destackification with restricted root operations

Abstract

e exhibit a modified version of Bergh’s functorial destackification algorithm which employs mth root operations only for positive integers $m$ not divisible by a given prime $p$ when applied to stacks whose geometric stabilizer group schemes do not contain pth roots of unity. When $p$ is the characteristic of a given base field, this has the consequence that it is possible to destackify a tame Deligne–Mumford stack over the base field staying entirely within the realm of tame Deligne–Mumford stacks, rather than requiring more general tame Artin stacks. The modifications are based on two observations. First, destackification is possible for stacks with 3-torsion diagonalizable stabilizers, as for 2-torsion stabilizers, with just ordinary blow-ups, and not the more general stacky blow-ups that enlarge stabilizer groups. Second, modulo a given prime greater than or equal to 5, every nonzero residue class can be expressed as a product of smaller primes different from p.