Destackification with restricted root operations
type: Item
instance of: scholarly article
title: Destackification with restricted root operations
zbMATH Open document ID: 1408.14007
author: Andrew Kresch
published in: European Journal of Mathematics
publication date: 2018-12-07T00:00:00Z
full work available at URL: kresch destackification.pdf
review text: Let \(X\) be a smooth separated Deligne-Mumford stack of finite type over a field \(k\). The stack \(X\) is assumed to be \textit{tame}, i.e., \(\text{char}(k)\) does not divide the order of the stabilizer group at any geometric point of \(X\) which is equivalent to that \(X\) has the étale local structure over its coarse moduli space of a quotient stack \([U/\Gamma]\) with \(U\) a smooth affine \(k\)-scheme and \(\Gamma\) a finite group whose order is not divisible by the characteristic of \(k\). The article considers the functorial destackification of \textit{D. Bergh} [Compos. Math. 153, No. 6, 1257--1315 (2017; Zbl 1372.14002)]. This gives a way of replacing the stack \(X\) by a smooth stack \(\tilde X\) through a series of elementary operations such that the coarse moduli space of \(\tilde X\) is smooth and such that it may be reconstructed from its course moduli space by adding stabilizers over a normal crossing divisor. This reconstruction algorithm is not optimal: When \(\text{char}(k)=p>0,\) it is possible that the stabilizer group schemes \(\mu_m\) where \(m\) divides \(p\) arise, and then \(\tilde X\) no longer has the simple local form \([U/\Gamma].\) This says that the constructed stack no longer is a tame Deligne-Mumford stack, but belongs to the class of tame Artin stacks. This article gives simple modifications to the construction of \(\tilde X\) given by Bergh, staying within the class of tame Deligne-Mumford stacks. These kind of stacks are more easy to control as they belong to a more restricted class of stacks. Bergh's destackification procedure contain two kinds of operations: (i) Blowing up along a smooth center, and (ii) root stack along a smooth divisor. A root operation is obtained by attaching a positive integer \(m\) and a divisor, and replacing each stabilizer group by a \(\mu_m\)-extension along the divisor without altering the coarse moduli space. \textit{J. Oesinghaus} [``Conic bundles and iterated root stacks'', to appear in Eur. J. M…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 6990720
zbMATH Keywords: algebraic stack
zbMATH Keywords: destackification
zbMATH Keywords: root stack
zbMATH Keywords: tame stack
zbMATH Keywords: Artin stack
zbMATH Keywords: Deligne-Mumford stack
zbMATH Keywords: coarse moduli
zbMATH Keywords: stabilizer subgroup
zbMATH Keywords: divisorialification
zbMATH Keywords: admissible blowup
zbMATH Keywords: standard pair
MaRDI profile type: Publication
OpenAlex ID: W2888688066
cites work: Gromov-Witten theory of Deligne-Mumford stacks
cites work: Tame stacks in positive characteristic
cites work: Compactifying the space of stable maps
cites work: Functorial destackification of tame stacks with abelian stabilisers
cites work: The orbifold Chow ring of toric Deligne-Mumford stacks
cites work: Using stacks to impose tangency conditions on curves
cites work: Logarithmic geometry, minimal free resolutions and toric algebraic stacks
cites work: Conic bundles and iterated root stacks
Wikidata QID: Q129341309
DOI: 10.1007/S40879-018-0281-6
description: scientific article; zbMATH DE number 6990720