Destackification with restricted root operations
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. Math, \url{doi:10.1007/s40879-018-0251-z}] observed that when every geometric point of \(X\) has stabilizer \((\mu_2)^n\) for some \(n\geq 0\), Bergh's destackification algorithm yields a succession of blow-ups without nontrivial root operations. The author mention that stabilizers \((\mu_3)^n\) and destackification appears in the context of Brauer-Severi surface bundles. The article starts of with recalling Berghs procedure for destackification. Divisorialificaton produces a standard pair in which the stack is a gerbe over a tame Artin stack with diagonalizable stabilizers. This requires only the operation of blowing up. Consider a finite ordered set \(\mathbf E=(E^1,\dots,E^r)\) of effective Cartier divisors on \(X\), smooth over a quasi-compact scheme, which form a simple normal crossing divisor. After admissible blowup, replacing \(\mathbf E\), this gives a new distinguished structure \(\mathbf D=(E^{d+1},\dots, E^{r+1}).\) An essential part of the destackification algorithm is a \textit{partial toric destackification} of a combinatorial nature. This takes the form of an orbifold fan consisting of a lattice \(N\), a simplicial fan \(\Sigma\) whose support span \(N_{\mathbb R},\) and a homomorphism \(\beta\) from the free Abelian group with basis the set \(\Sigma(1)\) of rays in \(\Sigma\) to \(N,\) sending the generator corresponding to a given ray to a nonzero lattice point on the ray. In the destackification process, the following lemma is of independent interest: Let \((N,\Sigma,\beta)\) be an orbifold fan, whose rays are numbered and endowed with a distinguished structure. Then the following are equivalent: (i) All distinguished rays are independent. (ii) For every cone \(\rho\in\Sigma,\) with face \(\rho^{\text{nd}}\) generated by the non-distinguished rays of \(\rho,\) the homomorphism \(N_{\rho^{\text{nd}}}/N^1_{\rho^{\text{nd}}}\rightarrow N_{\rho}/N^1_\rho\) is an isomorphism. This result leads to the fact that if \(\sigma\in\Sigma\) satisfies an independence relation, then so does every face of \(\sigma.\) Now, we state two of the main results of the article verbatim: Theorem 4.1. Let \(p\) be a prime number and \(S\) a quasi-compact base scheme. Then there is a destackification algorithm for standard pairs \((X,\mathbf E)\) over \(S,\) functorial for change of base \(S^\prime\rightarrow S\) and gerbes and smooth stabilizer preserving morphisms \(X^\prime\rightarrow X,\) with the property that if the geometric stabilizer group schemes of \(X\) have no subgroup scheme isomorphic to \(\mu_p\), then this property is maintained by the destackification. Corollary 4.4. Let \(S\) be a quasi-compact base scheme and \((X,\mathbf E)\) a standart pair over \(S\) such that every geometric stabilizer group of \(X\) is isomorphic to \((\mu_3)^n\) for some natural number \(n\). Then the destackification algorithm of theorem 4.1 yields (for any \(p\)) a succesion of blow-ups without nontrivial root operations. The article gives a nice view on different techniques working with special classes of stacks. It explains in a relevant way the meaning of \textit{destackification}, a kind of removing the most stacky properties of an object by blowing up along divisors due to the ramification, and gives an improvement of Bergh's algorithm.
scientific article; zbMATH DE number 6990720
scientific article