# The singularity of the algebraic stack and the singularity of the coarse moduli space

It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse moduli space. Here I assume that the algebraic stack is locally of finite presentation, and has finite diagonal. More concretely.

1) Does the stack being normal imply that the coarse moduli space is normal?

2) Does the stack being log smooth (in the sense of logarithmic geometry) imply that the coarse moduli space is log smooth?

3) How about finite quotient singularities?

Thanks for the answers for a separated DM stacks. Since these stacks are locally finite quotients of algebraic spaces, 1) and 3) are positive, 2) is negative. However, what if the stack is not DM? I want to only assume that the stack is an Artin stack with finite diagonal.

• If $X={\rm{Spec}}(A)$ is a smooth affine variety of dimension $> 1$ over field $k$ and $G$ is a finite group acting on $X$ such that $X(k)^G$ is non-empty then $[X/G]$ is a $k$-smooth DM stack(since $X$ is a $k$-smooth finite etale cover) yet its coarse space is ${\rm{Spec}}(A^G)$, so (2) usually fails. More generally, if your stack $Y$ is DM and separated of finite type over a noetherian ring, so it has a coarse space $Y_0$ (say by Keel--Mori) then the henselian local rings on $Y_0$ are invariants of those on $Y$ under a finite group action, so (1) and (3) are OK in such cases. – user76758 May 2 '14 at 2:06
• I required dimension $> 1$ precisely because group actions on smooth curves exhibit much simpler behavior than in higher dimensions (e.g., passing to coarse quotient preserves smoothness, unlike nearly all cases beyond dimension 1, and the stack quotient is certainly smooth). Do I misunderstand your example? But (1) is true in great generality since finite group invariants preserves normality (I'm not sure why you think it fails). The failure of log-smoothness should occur beyond dimension 1 in char. 0 when $G$ is highly non-abelian, but for a rigorous proof ask a local expert in log geometry. – user76758 May 2 '14 at 3:21
• Sorry. I was stupid. You are right. On dimension 1, the quotient is smooth, it's certainly normal. Then I should ask for an example for 2). – user38276 May 2 '14 at 4:41

## 1 Answer

If $\mathcal{X}$ is a normal Deligne-Mumford stack then its coarse moduli space $X$ is normal. Since $\mathcal{X}$ its normal its admits an étale atlas $U_i\rightarrow\mathcal{X}$, with $U_i$ normal schemes. Now, the statement follows because $U_i$ is normal and $G$ is a finite group acting on $U_i$ then $U_i/G$ is normal as well. In dimension one this translate into the fact that a quotient of a smooth curve is smooth.

In particular, if $\mathcal{X}$ is a smooth Deligne-Mumford stack the coarse moduli space $X$ has finite quotient singularities, that is étale locally it is isomorphic to a quotient of a smooth scheme by a finite group, and $X$ is normal.

Logarithmic geometry fits well for instance with moduli spaces. Indeed $\overline{M}_{g,n}$ is a moduli space for log curves. See

Fumiharu Kato, "Log smooth deformation and moduli of log smooth curves", Internat. J. Math., 11(2), 215–232, 2000.

If $X$ is a smooth scheme, and $D\subset X$ is a divisor. We may a log structure on $X$ with respect to the divisor $D$ as $$M(U) = \{f\in \mathcal{O}_X(U) \; | \; f_{|U\setminus D}\in\mathcal{O}_{X}^{*}(U\setminus D)\}.$$ When $D$ is normal crossing it is log smooth. The answer to $(2)$ is negative. For instance consider $\overline{M}_{1,2}$. From the analysis of the singularities of $\overline{M}_{1,2}$ below you can see that $\mathcal{E}xt^1(\Omega_{\overline{M}_{1,2}},\mathcal{O}_{\overline{M}_{1,2}})$ is not trivial. Then $\overline{M}_{1,2}$ is not $d$-semistable.

As I wrote if $\mathcal{X}$ is a smooth Deligne-Mumford stack the coarse moduli space $X$ has finite quotient singularities. A nice example is the moduli space of pointed curves $\overline{M}_{g,n}$. If $g = 0$ the stack coincides with the coarse moduli space because $n$-pointed rational stable curves are automorphism-free. Anyway one can see the difference between the stack and the coarse moduli space already for $g = 1$.

It is well known that $\overline{M}_{1,1}\cong \mathbb{P}^{1}$ and $\overline{\mathcal{M}}_{1,1}\cong \mathbb{P}(4,6)$. Clearly $\mathbb{P}^{1}\cong \mathbb{P}(4,6)$ as varieties. However they are not isomorphic as stacks. Indeed $\mathbb{P}(4,6)$ has two stacky points with stabilizers $\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$. These two points are indistinguishable from any other point on the coarse moduli space $\overline{M}_{1,1}$.

For singularities we have to look at the case $g=1, n=2$. Since $\overline{\mathcal{M}}_{1,2}$ is a smooth Deligne-Mumford stack the coarse moduli space $\overline{M}_{1,2}$ will have finite quotient singularities at the places where the automorphism groups jump.

Indeed $\overline{M}_{1,2}$ is a rational surface with four singular points. Two singular points lie in $M_{1,2}$, and are:

• a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve with automorphism group $\mathbb{Z}_4$ and marked points $[0:1:0]$ and $[0:0:1]$;
• a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve with automorphism group $\mathbb{Z}_6$, and marked points $[0:1:0]$ and $[0:1:1]$.

The remaining two singular points lie on the boundary divisor, and are:

• a singularity of type $\frac{1}{6}(2,4)$ representing a reducible curve whose irreducible components are an elliptic curve with automorphism group $\mathbb{Z}_6$ and a smooth rational curve connected by a node;
• a singularity of type $\frac{1}{4}(2,6)$ representing a reducible curve whose irreducible components are an elliptic curve with automorphism group $\mathbb{Z}_4$ and a smooth rational curve connected by a node.

For other manifestations of these phenomena you may look at stacks of weighted pointed curves, prym, spin, and level curves (http://arxiv.org/abs/1205.0201).

Remark: Note that a singular stack could have a smooth coarse moduli space. For instance, take $X = L\cup R\subset\mathbb{A}^2$ be the union of two lines. We have an action of $S_2$ on $X$ switching $L$ and $R$. Now, the quotient stack is singular because it admits an étale cover by something singular. However, the coarse moduli space is $\mathbb{A}^1$.

• Thank you very much. Especially for the detailed list of singularities on $\overline{M}_{1,2}$. – user38276 May 2 '14 at 16:38