Partial desingularizations of good moduli spaces of Artin toric stacks

Abstract

Let X be an Artin stack with good moduli space X → M . We define the Reichstein transform of X relative to a closed substack C ⊂ X to be the complement of the strict transform of the saturation of C in the blowup of X along C. The main technical result of the paper is that the Reichstein transform of a toric Artin stack relative to a toric substack is again a toric stack. Precisely, the Reichstein transform relative to a cone in a stacky fan is the toric stack determined by stacky star subdivision. This leads to our main theorem which states that for toric Artin stacks there is a canonical sequence of Reichstein transforms that produces a toric Deligne-Mumford stack. When the good moduli space of the toric stack is a projective toric variety our procedure can be interpreted in terms of Kirwan's [Kir] partial desingularization of geometric invariant theory quotients.

Functorial destackification of tame stacks with abelian stabilisers

“…We define what is meant by the star subdivision of a stacky fan. This is the same definition as made by Edidin in [EM12].…”

“…For smooth toric varieties, blow-ups at orbit closures correspond to star subdivisions. This generalises to toric orbifolds (see [EM12]). We recall the definition here.…”

Comparing the Kirwan and noncommutative resolutions of quotient varieties

“…The steps in the partial resolution of X/ /G described in [Kir85] were reinterpreted by Reichstein [Rei89], and generalized by Edidin and More [EM12] and Edidin and Rydh [ER17]. They are now known as "Reichstein transforms" [EM12]. We will use (−) R for notations related to the Reichstein transform.…”

Integration on Artin toric stacks and Euler characteristics

“…The following result gives expresses how the Euler characteristic increases after a stacky star subdivision [EM,Definition 4.1] of a simplicial stacky fan.…”

“…Let Σ be a (not necessarily simplicial) stacky fan and let X (Σ) be the associated Artin toric stack. By [EM,Theorem 5.2] there is a simplicial stacky fan Σ ′ canonically obtained from Σ by stacky star subdivisions and a commutative diagram of stacks and toric varieties…”