13
$\begingroup$

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism away from codimension 2?

If $X$ is finite type over a field, the answer is yes, by the following argument. Theorem 4.6 of Fantechi-Mann-Nironi's Smooth toric DM stacks shows that such stacks have a universal property, so it suffices to show that they exist étale-locally (the universal property ensures the locally constructed canonical stacks will glue). Étale locally, we may assume $X=U/G$, where $U$ is smooth and $G$ is a finite group acting tamely on $U$—that's the definition of "tame quotient singularities". Let $H\subseteq G$ be the (normal) subgroup acting by pseudoreflections (through a given closed point $x\in U$; shrinking $U$ if necessary). Then by the Chevalley-Shephard-Todd theorem, $T_x/H$ is smooth, where $T_x$ is the tangent space at $x\in U$. If $X$ is defined over the residue field $k(x)$, then we can construct an $H$-equivariant morphism $U\to T_x$ sending $x$ to the origin, which is étale at $x$. Since $T_x/H$ is smooth, so is $U/H$. (If $X$ isn't defined over $k(x)$, base change to $k(x)$ and get smoothness of $U/H$ by descent.) The action of $G/H$ on $U/H$ is free away from codimension 2, so $\mathcal X = [(U/H)/(G/H)]$ does the trick.

As far as I can tell, the only place we had to use that $X$ is defined over a field was in showing smoothness of $U/H$. Without it, we can't construct the étale morphism $U/H\to T_x/H$ we need to deduce smoothness of $U/H$ from smoothness of $T_x/H$.

Can this "over a field" condition be relaxed to get the existence of canonical stacks in an absolute setting (i.e. over $\mathrm{Spec}(\mathbb Z)$)?

Context: my more general goal is to understand if canonical stacks exist over an arbitrary base $S$. That is, suppose $X$ is a scheme (probably locally of finite type) over a base $S$ which has an étale cover by a disjoint union of schemes of the form $U/G$, where $U$ is smooth over $S$, and $G$ is an abstract group acting on $U$ (over $S$), with order relatively prime to all residue fields of $U$. Then does there exist a stack $\mathcal X$ which is smooth over $S$ and has coarse space $X$, such that the coarse space morphism is an isomorphism away from codimension 2 (on both $X$ and $\mathcal X$)?

$\endgroup$
7
$\begingroup$

In the smooth case, I think that the answer is positive over an arbitrary regular excellent base. The argument was in my PhD thesis; it was done over a field, but I think it adapts to this case.

Cover your $X$ in the étale topology with schemes of the form $U/G$, where $G$ is prime to all the residue characteristics of $U$, and $U$ is smooth over $S$. Choose a geometric point $p$ of $U$; we can restrict to the stabilizer of $G$, and assume that $G$ leaves $p$ fixed. Call $H$ the subgroup generated by the elements of $G$ that are pseudoreflections, or the identity, when restricted to the fiber along $p$; this is normal in $G$. The scheme $U/H$ is flat over $S$, because of the tameness hypothesis. Furthermore taking quotients by $H$ commutes with base change on $S$, again because of tameness; hence the geometric fiber of $U/H \to S$ along the image of $p$ is the quotient of the geometric fiber, which is smooth, because of Cartan’s and Chevalley’s theorems. In this way we can assume that $G$ stabilizes $p$, and the restriction of $G$ to the fiber through $p$ contains no pseudoreflexions.

Now look at the locus in which $U \to X$ is étale. Notice that if the restriction of $U \to X$ to the fiber is étale at $p$, then $U \to X$ is étale at $p$, by the local criterion of flatness. The locus on which $U \to X$ is étale is open in X; its complement must have codimension larger than $1$, because otherwise it would intersect the fiber in codimension $1$.

Now take two of these charts $U \to X$ and $V \to X$; the normalization of the part of the fibered product $U \times_X V$ that dominates $X$ is étale over $U$ and $V$, by purity of the branch locus. These data give a Q-variety, in the sense of Mumford; from them you get an étale groupoid that defines the stack that you are looking for.

When S is not regular, I am really not sure, I suspect it might be false.

$\endgroup$
  • $\begingroup$ Thanks. Could you explain how the "regular" and "excellent" hypotheses on $S$ are being used? I think excellence of $S$ is needed to get the étale locus of $U\to X$ to be open and of codimension 1. Is regularity of $S$ needed to get $U/H$ to be smooth over $S$ at $p$? $\endgroup$ – Anton Geraschenko Feb 8 '15 at 20:22
  • $\begingroup$ No, I don’t think that $S$ being regular is necessary to conclude that $U/H$ is smooth; flatness follows from the fact that $H$ is tame, and then smoothness can be checked on the fibers. The fact that $S$ is regular is used in the last step, when you normalize the fiber product $U \times_S V$, and use purity of the branch locus to conclude that it is étale over$U$ and $V$. $\endgroup$ – Angelo Feb 8 '15 at 21:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.