1 Introduction
This is the second paper in a trilogy in which we construct the second flip in the log minimal model program for $\overline{M}_{g,n}$ . In this paper, we prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$ -stable curves introduced in [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, § 2] admit good moduli spaces. Namely, we prove the following result.
Theorem 1.1. For every $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},1]$ , $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ admits a good moduli space $\overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC})$ which is a proper algebraic space over $\mathbb{C}$ . Furthermore, for each critical value $\unicode[STIX]{x1D6FC}_{c}\in \{2/3,7/10,9/11\}$ , there exists a diagram
where $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})\rightarrow \overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ , $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}+\unicode[STIX]{x1D716})\rightarrow \overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}+\unicode[STIX]{x1D716})$ , and $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}-\unicode[STIX]{x1D716})\rightarrow \overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}-\unicode[STIX]{x1D716})$ are good moduli spaces, and where $\overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}+\unicode[STIX]{x1D716})\rightarrow \overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ and $\overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}-\unicode[STIX]{x1D716})\rightarrow \overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ are proper morphisms of algebraic spaces.
The goal of the final paper in the trilogy [Reference Alper, Fedorchuk and SmythAFS16] is to establish an isomorphism between $\overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC})$ and the projective variety
which thereby proves that the good moduli spaces of $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ are indeed the log canonical models defined in (1.1).
In the case where $\unicode[STIX]{x1D6FC}\in (2/3,7/10]$ and $n=0$ , the spaces $\overline{\mathbb{M}}_{g,n}(\unicode[STIX]{x1D6FC})$ have been constructed using geometric invariant theory (GIT) by Hassett and Hyeon in [Reference Hassett and HyeonHH09, Reference Hassett and HyeonHH13]. There is no known GIT construction of the moduli spaces corresponding to $\unicode[STIX]{x1D6FC}\leqslant 2/3$ . Thus, this paper gives the first intrinsic construction of a moduli space associated to an algebraic stack parameterizing objects with infinite automorphism groups.
In order to prove Theorem 1.1, we establish three general existence results for good moduli spaces. These existence results make essential use of the notions of local quotient presentation and variation of geometric invariant theory (VGIT) chambers of a local quotient presentation, introduced in Definitions 2.1 and 2.4, respectively. Our first existence result gives conditions under which one may use a local quotient presentation to construct a good moduli space.
Theorem 1.2. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ . Suppose that:
-
(1) for every closed point $x\in {\mathcal{X}}$ , there exists a local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ around $x$ such that:
-
(a) the morphism $f$ is stabilizer preserving at closed points of ${\mathcal{W}}$ ; and
-
(b) the morphism $f$ sends closed points to closed points; and
-
-
(2) for any point $x\in {\mathcal{X}}(k)$ , the closed substack $\overline{\{x\}}$ admits a good moduli space.
Then ${\mathcal{X}}$ admits a good moduli space.
As we explain below, this result may be considered as an analog of the Keel–Mori theorem [Reference Keel and MoriKM97] for algebraic stacks, but in practice the hypotheses of Theorem 1.2 are harder to verify than those of the Keel–Mori theorem. Nevertheless, we believe that the above theorem should be applicable to many additional moduli problems. In fact, it has come to our attention that it has already been applied to construct a good moduli space of Kähler–Einstein Fano varieties in [Reference Li, Wang and XuLWX14, Reference OdakaOda15].
Our second existence result, Theorem 1.3, gives one situation in which the hypotheses of Theorem 1.2 are satisfied. It says that if ${\mathcal{X}}$ is an algebraic stack and ${\mathcal{X}}^{+}{\hookrightarrow}{\mathcal{X}}{\hookleftarrow}{\mathcal{X}}^{-}$ is a pair of open immersions locally cut out by VGIT chambers of a local quotient presentation, then ${\mathcal{X}}$ admits a good moduli space if ${\mathcal{X}}^{+}$ , ${\mathcal{X}}\smallsetminus {\mathcal{X}}^{+}$ , and ${\mathcal{X}}\smallsetminus {\mathcal{X}}^{-}$ do.
Theorem 1.3. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ and let ${\mathcal{L}}$ be a line bundle on ${\mathcal{X}}$ . Let ${\mathcal{X}}^{+},{\mathcal{X}}^{-}\subset {\mathcal{X}}$ be open substacks, and let ${\mathcal{Z}}^{+}={\mathcal{X}}\smallsetminus {\mathcal{X}}^{+}$ , ${\mathcal{Z}}^{-}={\mathcal{X}}\smallsetminus {\mathcal{X}}^{-}$ be their reduced complements. Suppose that:
-
(i) the algebraic stacks ${\mathcal{X}}^{+}$ , ${\mathcal{Z}}^{+}$ , ${\mathcal{Z}}^{-}$ admit good moduli spaces; and
-
(ii) for all closed points $x\in {\mathcal{Z}}^{+}\cap {\mathcal{Z}}^{-}$ , there exist a local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ around $x$ and a Cartesian diagram
(1.2)where ${\mathcal{W}}_{{\mathcal{L}}}^{+},{\mathcal{W}}_{{\mathcal{L}}}^{-}$ are the VGIT chambers of ${\mathcal{W}}$ with respect to ${\mathcal{L}}$ .
Then there exist good moduli spaces ${\mathcal{X}}\rightarrow X$ and ${\mathcal{X}}^{-}\rightarrow X^{-}$ such that $X^{+}\rightarrow X$ and $X^{-}\rightarrow X$ are proper and surjective. In particular, if $X^{+}$ is proper over $k$ , then $X$ and $X^{-}$ are also proper over $k$ .
The third existence result, Proposition 1.4, proves that one can check existence of a good moduli space after passing to a finite cover by a quotient stack. Recall that an algebraic stack ${\mathcal{X}}$ is called a global quotient stack if ${\mathcal{Z}}\simeq [Z/\text{GL}_{n}]$ , where $Z$ is an algebraic space with an action of $\operatorname{GL}_{n}$ .
Proposition 1.4. Let $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ be a morphism of algebraic stacks of finite type over an algebraically closed field $k$ of characteristic $0$ . Suppose that:
-
(i) the morphism $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ is finite and surjective;
-
(ii) there exists a good moduli space ${\mathcal{X}}\rightarrow X$ with $X$ separated; and
-
(iii) the algebraic stack ${\mathcal{Y}}$ is a global quotient stack and admits local quotient presentations.
Then there exists a good moduli space ${\mathcal{Y}}\rightarrow Y$ with $Y$ separated. Moreover, if $X$ is proper, so is $Y$ .
Both Theorem 1.3 and Proposition 1.4 are proved using Theorem 1.2.
1.1 Motivation and sketch of proof of Theorem 1.2
In order to motivate the statement of Theorem 1.2, let us give an informal sketch of the proof. If ${\mathcal{X}}$ admits local quotient presentations, then every closed point $x\in {\mathcal{X}}$ admits an étale and affine neighborhood of the form
where $A_{x}$ is a $k$ -algebra of finite type and $G_{x}$ is the stabilizer of $x$ . The union
defines an étale cover of ${\mathcal{X}}$ ; reducing to a finite subcover, we obtain an atlas $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ with the following properties:
-
(i) $f$ is étale and affine; and
-
(ii) ${\mathcal{W}}$ admits a good moduli space $W$ .
Indeed, (ii) follows simply by taking invariants $[\operatorname{Spec}A_{x}/G_{x}]\rightarrow \operatorname{Spec}A_{x}^{G_{x}}$ . Since $f$ is affine, the fiber product ${\mathcal{R}}:={\mathcal{W}}\times _{{\mathcal{X}}}{\mathcal{W}}$ admits a good moduli space $R$ . We may thus consider the following diagram:
The crucial question is: can we choose $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ to guarantee that the projections $q_{1},q_{2}:R\rightrightarrows W$ define an étale equivalence relation? If so, then the algebraic space quotient $X=W/R$ gives a good moduli space for ${\mathcal{X}}$ .
If ${\mathcal{X}}$ is a separated Deligne–Mumford stack, we can always do this. Indeed, the atlas $f$ may be chosen to be stabilizer preserving.Footnote 1 Thus, we may take the projections ${\mathcal{R}}\rightrightarrows {\mathcal{W}}$ to be stabilizer preserving and étale, and this implies that the projections $R\rightrightarrows W$ are étale.Footnote 2 This leads to a direct proof of the Keel–Mori theorem for separated Deligne–Mumford stacks of finite type over $\operatorname{Spec}k$ (one can show directly that such stacks always admit local quotient presentations). In general, of course, algebraic stacks need not be separated so we must find weaker conditions which ensure that the projections $q_{1},q_{2}$ are étale. In particular, we must identify a set of sufficient conditions that can be directly verified for geometrically defined stacks such as $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ .
Our result gives at least one plausible answer to this problem. To begin, note that if $\unicode[STIX]{x1D714}\in {\mathcal{W}}$ is a closed $k$ -point with image $w\in W$ , then the formal neighborhood $\widehat{{\mathcal{O}}}_{W,w}$ can be identified with the $G_{\unicode[STIX]{x1D714}}$ -invariants $D_{\unicode[STIX]{x1D714}}^{G_{\unicode[STIX]{x1D714}}}$ of the miniversal deformation space $D_{\unicode[STIX]{x1D714}}$ of $\unicode[STIX]{x1D714}$ . Thus, we may ensure that $q_{i}$ is étale at a $k$ -point $r\in R$ , or equivalently that the induced map $\widehat{{\mathcal{O}}}_{W,q_{i}(r)}\rightarrow \widehat{{\mathcal{O}}}_{R,r}$ is an isomorphism, by manually imposing the following conditions: $p_{i}(\unicode[STIX]{x1D70C})$ should be a closed point, where $\unicode[STIX]{x1D70C}\in {\mathcal{R}}$ is the unique closed point in the preimage of $r\in R$ , and $p_{i}$ should induce an isomorphism of stabilizer groups $G_{\unicode[STIX]{x1D70C}}\simeq G_{p_{i}(\unicode[STIX]{x1D70C})}$ . Indeed, we then have $\widehat{{\mathcal{O}}}_{W,q_{i}(r)}=D_{p_{i}(\unicode[STIX]{x1D70C})}^{G_{p_{i}(\unicode[STIX]{x1D70C})}}\simeq D_{\unicode[STIX]{x1D70C}}^{G_{\unicode[STIX]{x1D70C}}}=\widehat{{\mathcal{O}}}_{R,r}$ , where the middle isomorphism follows from the hypothesis that $p_{i}$ is étale and stabilizer preserving. In sum, we have identified two key conditions that will imply that $R\rightrightarrows W$ is an étale equivalence relation:
- $(\star )$
-
the morphism $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ is stabilizer preserving at closed points;
- $(\star \star )$
-
the projections $p_{1},p_{2}:{\mathcal{W}}\times _{{\mathcal{X}}}{\mathcal{W}}\rightrightarrows {\mathcal{W}}$ send closed points to closed points.
Condition ( $\star$ ) is precisely hypothesis (1a) of Theorem 1.2. In practice, it is difficult to directly verify condition ( $\star \star$ ), but it turns out that it is implied by conditions (1b) and (2) of Theorem 1.2, which are often easier to verify.
1.2 Roadmap
In § 2, we recall the necessary terminology and results from [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17]. We also prove some preliminary lemmas concerning strongly étale morphisms. In § 3, we prove Theorems 1.2 and 1.3 and Proposition 1.4. Namely, in § 3.1, we prove Theorem 1.2 along the lines described in § 1.1. Then we prove Theorem 1.3 and Proposition 1.4 in §§ 3.2 and 3.3, respectively, by showing that after suitable reductions, their hypotheses imply that conditions (1a), (1b), and (2) of Theorem 1.2 are satisfied. In § 4, we apply Theorem 1.3 to prove Theorem 1.1. Appendix A provides various examples of algebraic stacks highlighting the necessity of conditions (1a), (1b), and (2) in Theorem 1.2.
Notation
Theorem 1.1 holds over an arbitrary algebraically closed field $\mathbb{C}$ of characteristic $0$ . In §§ 2 and 3, we work over an algebraically closed field $k$ of arbitrary characteristic unless when stated otherwise. A linearly reductive group scheme over a field $k$ is, by definition, an affine group scheme of finite type over $k$ such that every representation is completely reducible.
2 Background and preliminary results
2.1 Local quotient presentations
Definition 2.1. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ and let $x\in {\mathcal{X}}(k)$ be a closed point. We say that $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ is a local quotient presentation around $x$ if:
-
(i) the stabilizer $G_{x}$ of $x$ is linearly reductive;
-
(ii) there is an isomorphism ${\mathcal{W}}\simeq [\operatorname{Spec}A/G_{x}]$ , where $A$ is a $k$ -algebra of finite type;
-
(iii) the morphism $f$ is étale and affine; and
-
(iv) there exists a point $w\in {\mathcal{W}}(k)$ such that $f(w)=x$ and $f$ induces an isomorphism of stabilizer group schemes at $x$ .
We sometimes write $f:({\mathcal{W}},w)\rightarrow ({\mathcal{X}},x)$ as a local quotient presentation to indicate the chosen preimage of $x$ . We say that ${\mathcal{X}}$ admits local quotient presentations if there exist local quotient presentations around all closed points $x\in {\mathcal{X}}(k)$ .
Remark 2.2. Note that if ${\mathcal{X}}$ admits local quotient presentations, then ${\mathcal{X}}$ necessarily has affine diagonal and every closed point necessarily has a linearly reductive stabilizer group.
For our purpose of applying Theorem 1.3 to the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ , the following result suffices to guarantee the existence of local quotient presentations.
Proposition 2.3 [Reference Alper and KreschAK16, § 3.3].
Let $k$ be an algebraically closed field. Let ${\mathcal{X}}$ be a quotient stack $[U/G]$ , where $U$ is a normal and separated scheme of finite type over $k$ and $G$ is a smooth linear algebraic group over $k$ . If $x\in {\mathcal{X}}(k)$ is a point with linearly reductive stabilizer, then there exists a local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ around $x$ .
2.2 Local VGIT chambers
Let $G$ be a linearly reductive group scheme over an algebraically closed field $k$ . Let $X$ be a $k$ -scheme with an action of $G$ via $\unicode[STIX]{x1D70E}:G\times X\rightarrow X$ . Let $\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}=\operatorname{Spec}k[t]_{t}$ be a character. Set $A_{n}:=\{f\in A\mid \unicode[STIX]{x1D70E}^{\ast }(f)=\unicode[STIX]{x1D712}^{\ast }(t)^{-n}f\}$ . We define the VGIT ideals with respect to $\unicode[STIX]{x1D712}$ to be
The VGIT $(+)$ -chamber and $(-)$ -chamber of $X$ with respect to $\unicode[STIX]{x1D712}$ are the open subschemes
and, since the open subschemes $X_{\unicode[STIX]{x1D712}}^{+}$ , $X_{\unicode[STIX]{x1D712}}^{-}$ are $G$ -invariant, we also have stack-theoretic open immersions
We will refer to these open immersions as the VGIT $(+)/(-)$ -chambers of $[X/G]$ with respect to $\unicode[STIX]{x1D712}$ .
Finally, given the data of an algebraic stack ${\mathcal{X}}$ and a line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ , we can define the VGIT chambers of a local quotient presentation. In this situation, note that if $x\in {\mathcal{X}}(k)$ is any point, then there is a natural action of the automorphism group $G_{x}$ on the fiber ${\mathcal{L}}|_{BG_{x}}$ that induces a character $\unicode[STIX]{x1D712}_{{\mathcal{L}}}:G_{x}\rightarrow \mathbb{G}_{m}$ .
Definition 2.4 (VGIT chambers of a local quotient presentation).
Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ . Let ${\mathcal{L}}$ be a line bundle on ${\mathcal{X}}$ . Let $x\in {\mathcal{X}}$ be a closed point. If $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ is a local quotient presentation around $x$ , we define the chambers of ${\mathcal{W}}$ with respect to ${\mathcal{L}}$ to be the VGIT $(+)/(-)$ -chambers
of ${\mathcal{W}}$ with respect to the character $\unicode[STIX]{x1D712}_{{\mathcal{L}}}:G_{x}\rightarrow \mathbb{G}_{m}$ .
2.3 Strongly étale morphisms
Definition 2.5. Let $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ be a morphism of algebraic stacks of finite type over an algebraically closed field $k$ . Assume that ${\mathcal{X}}$ and ${\mathcal{Y}}$ have affine diagonal. We say that:
-
– the morphism $f$ sends closed points to closed points if for every closed point $x\in {\mathcal{X}}$ , the point $f(x)\in {\mathcal{Y}}$ is closed;
-
– the morphism $f$ is stabilizer preserving at $x\in {\mathcal{X}}(k)$ if $\operatorname{Aut}_{{\mathcal{X}}(k)}(x)\rightarrow \operatorname{Aut}_{{\mathcal{Y}}(k)}(f(x))$ is an isomorphism;
-
– for a closed point $x\in {\mathcal{X}}$ , the morphism $f$ is strongly étale at $x$ if $f$ is étale at $x$ , $f$ is stabilizer preserving at $x$ , and $f(x)\in {\mathcal{Y}}$ is closed;
-
– the morphism $f$ is strongly étale if $f$ is strongly étale at all closed points of ${\mathcal{X}}$ .
Definition 2.6. Let $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ be a good moduli space. We say that an open substack ${\mathcal{U}}\subset {\mathcal{X}}$ is saturated if $\unicode[STIX]{x1D719}^{-1}(\unicode[STIX]{x1D719}({\mathcal{U}}))={\mathcal{U}}$ .
The following proposition is simply a stack-theoretic formulation of Luna’s well-known results in invariant theory [Reference LunaLun73, Chapitre II] often referred to as Luna’s fundamental lemma. It justifies the terminology strongly étale by showing that strongly étale morphisms induce étale morphisms of good moduli spaces. It also shows that for a morphism of algebraic stacks admitting good moduli spaces, strong étaleness is an open condition.
Proposition 2.7. Let ${\mathcal{W}}$ and ${\mathcal{X}}$ be algebraic stacks of finite type over an algebraically closed field $k$ . Assume that ${\mathcal{W}}$ and ${\mathcal{X}}$ have affine diagonal. Consider a commutative diagram
where $f$ is representable and separated, and both $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D719}$ are good moduli spaces. Then:
-
(i) if $f$ is strongly étale at $w\in {\mathcal{W}}$ , then $g$ is étale at $\unicode[STIX]{x1D711}(w)$ ;
-
(ii) if $f$ is strongly étale, then $g$ is étale and Diagram (2.1) is Cartesian; and
-
(iii) there exists a saturated open substack ${\mathcal{U}}\subset {\mathcal{W}}$ such that:
-
(a) the morphism $f|_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow {\mathcal{X}}$ is strongly étale and $f({\mathcal{U}})\subset {\mathcal{X}}$ is saturated; and
-
(b) if $w\in {\mathcal{W}}$ is a closed point such that $f$ is strongly étale at $w$ , then $w\in {\mathcal{U}}$ .
-
Proof. Alper [Reference AlperAlp13, Theorem 5.1] gives part (i) and that $g$ is étale in (ii). The hypotheses in (ii) imply that the induced morphism $\unicode[STIX]{x1D6F9}:{\mathcal{W}}\rightarrow W\times _{X}{\mathcal{X}}$ is representable, separated, quasi-finite, and sends closed points to closed points. Alper [Reference AlperAlp13, Proposition 6.4] implies that $\unicode[STIX]{x1D6F9}$ is finite. Moreover, since $f$ and $g$ are étale, so is $\unicode[STIX]{x1D6F9}$ . But since ${\mathcal{W}}$ and $W\times _{X}{\mathcal{X}}$ both have $W$ as a good moduli space, it follows that a closed point in $W\times _{X}{\mathcal{X}}$ has a unique preimage under $\unicode[STIX]{x1D6F9}$ . Therefore, $\unicode[STIX]{x1D6F9}$ is an isomorphism and the diagram is Cartesian. Statement (iii) follows from [Reference AlperAlp10, Theorem 6.10].◻
Lemma 2.8. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ . Suppose that ${\mathcal{X}}$ has affine diagonal and that there exists a good moduli space $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ . Let $x\in {\mathcal{X}}$ be a closed point and ${\mathcal{U}}\subset {\mathcal{X}}$ be an open substack containing $x$ . Then there exists a saturated open substack ${\mathcal{U}}_{1}\subset {\mathcal{U}}$ containing $x$ . Moreover, if ${\mathcal{X}}\simeq [\operatorname{Spec}A/G]$ with $G$ linearly reductive, then ${\mathcal{U}}_{1}$ can be chosen to be of the form $[\operatorname{Spec}B/G]$ for a $G$ -invariant open affine subscheme $\operatorname{Spec}B\subset \operatorname{Spec}A$ .
Proof. The substacks $\{x\}$ and ${\mathcal{X}}\setminus {\mathcal{U}}$ are closed and disjoint. By [Reference AlperAlp13, Theorem 4.16], $\unicode[STIX]{x1D719}(\{x\})$ and $Z:=\unicode[STIX]{x1D719}({\mathcal{X}}\setminus {\mathcal{U}})$ are closed and disjoint. For the first statement, take ${\mathcal{U}}_{1}=\unicode[STIX]{x1D719}^{-1}(X\setminus Z)$ . For the second statement, take ${\mathcal{U}}_{1}=\unicode[STIX]{x1D719}^{-1}(U_{1})$ for an affine open subscheme $U_{1}\subset X\setminus Z$ .◻
Lemma 2.9. Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be algebraic stacks of finite type over an algebraically closed field $k$ . Suppose that ${\mathcal{X}}$ and ${\mathcal{Y}}$ have affine diagonal. Let $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ be a strongly étale morphism. Suppose that for points $x\in {\mathcal{X}}(k)$ and $y\in {\mathcal{Y}}(k)$ , the closed substacks $\overline{\{x\}}\subset {\mathcal{X}}$ and $\overline{\{y\}}\subset {\mathcal{Y}}$ admit good moduli spaces. Then, for any morphism of finite type $g:{\mathcal{Y}}^{\prime }\rightarrow {\mathcal{Y}}$ with affine diagonal, the base change $f^{\prime }:{\mathcal{X}}\times _{{\mathcal{Y}}}{\mathcal{Y}}^{\prime }\rightarrow {\mathcal{Y}}^{\prime }$ is strongly étale.
Proof. Clearly, the morphism $f^{\prime }$ is étale. Let $x^{\prime }\in {\mathcal{X}}\times _{{\mathcal{Y}}}{\mathcal{Y}}^{\prime }$ be a closed point. To check that $f^{\prime }$ is stabilizer preserving at $x^{\prime }$ and $f^{\prime }(x^{\prime })\in {\mathcal{Y}}^{\prime }$ is closed, we may replace ${\mathcal{Y}}$ with $\overline{\{g(f^{\prime }(x^{\prime }))\}}$ and ${\mathcal{X}}$ with $\overline{\{g^{\prime }(x^{\prime })\}}$ , where $g^{\prime }:{\mathcal{X}}\times _{{\mathcal{Y}}}{\mathcal{Y}}^{\prime }\rightarrow {\mathcal{X}}$ . Since $f$ is strongly étale, Proposition 2.7(ii) implies that $f$ is in fact an isomorphism, and the desired properties of $f^{\prime }$ are immediate.◻
3 General existence results
In this section, we prove Theorems 1.2 and 1.3 and Proposition 1.4.
3.1 Existence via local quotient presentations
Proposition 3.1. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ . Suppose that ${\mathcal{X}}$ has affine diagonal. Suppose that:
-
(i) there exists an affine, strongly étale, and surjective morphism $f:{\mathcal{X}}_{1}\rightarrow {\mathcal{X}}$ from an algebraic stack ${\mathcal{X}}_{1}$ admitting a good moduli space $\unicode[STIX]{x1D719}_{1}:{\mathcal{X}}_{1}\rightarrow X_{1}$ ; and
-
(ii) for any point $x\in {\mathcal{X}}(k)$ , the closed substack $\overline{\{x\}}$ admits a good moduli space.
Then ${\mathcal{X}}$ admits a good moduli space $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ .
Proof. Set ${\mathcal{X}}_{2}={\mathcal{X}}_{1}\times _{{\mathcal{X}}}{\mathcal{X}}_{1}$ . By Lemma 2.9, the projections $p_{1},p_{2}:{\mathcal{X}}_{2}\rightarrow {\mathcal{X}}_{1}$ are strongly étale. As $f$ is affine, there exists a good moduli space $\unicode[STIX]{x1D719}_{2}:{\mathcal{X}}_{2}\rightarrow X_{2}$ with projections $q_{1},q_{2}:X_{2}\rightarrow X_{1}$ . Similarly, the algebraic stack ${\mathcal{X}}_{3}:={\mathcal{X}}_{1}\times _{{\mathcal{X}}}{\mathcal{X}}_{1}\times _{{\mathcal{X}}}{\mathcal{X}}_{1}$ admits a good moduli space $\unicode[STIX]{x1D719}_{3}:{\mathcal{X}}_{3}\rightarrow X_{3}$ and the three projections ${\mathcal{X}}_{3}\rightarrow {\mathcal{X}}_{2}$ are strongly étale. By Proposition 2.7(ii), the induced diagram
is Cartesian. Moreover, by the universality of good moduli spaces, there are an induced identity map $X_{1}\rightarrow X_{2}$ , an inverse $X_{2}\rightarrow X_{2}$ , and a composition $X_{2}\times _{q_{1},X_{1},q_{2}}X_{2}\rightarrow X_{2}$ giving $X_{2}\rightrightarrows X_{1}$ an étale groupoid structure.
To check that $(q_{1},q_{2}):X_{2}\rightarrow X_{1}\times X_{1}$ is a monomorphism, it suffices to check that there is a unique preimage of $(x_{1},x_{1})\in X_{1}\times X_{1}$ , where $x_{1}\in X_{1}(k)$ . Let $\unicode[STIX]{x1D709}_{1}\in {\mathcal{X}}_{1}$ be the unique closed point in $\unicode[STIX]{x1D719}_{1}^{-1}(x_{1})$ . Since ${\mathcal{X}}_{1}\rightarrow {\mathcal{X}}$ is stabilizer preserving at $\unicode[STIX]{x1D709}_{1}$ , we can set $G:=\operatorname{Aut}_{{\mathcal{X}}_{1}(k)}(\unicode[STIX]{x1D709}_{1})\simeq \operatorname{Aut}_{{\mathcal{X}}(k)}(f(\unicode[STIX]{x1D709}_{1}))$ . There are diagrams
where the squares in the left-hand diagram are Cartesian. Suppose that $x_{2}\in X_{2}(k)$ is a preimage of $(x_{1},x_{1})$ under $X_{2}\rightarrow X_{1}\times X_{1}$ . Let $\unicode[STIX]{x1D709}_{2}\in {\mathcal{X}}_{2}$ be the unique closed point in $\unicode[STIX]{x1D719}_{2}^{-1}(x_{2})$ . Then $(p_{1}(\unicode[STIX]{x1D709}_{2}),p_{2}(\unicode[STIX]{x1D709}_{2}))\in {\mathcal{X}}_{1}\times {\mathcal{X}}_{1}$ is closed and is therefore the unique closed point $(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{1})$ in $(\unicode[STIX]{x1D719}_{1}\times \unicode[STIX]{x1D719}_{1})^{-1}(x_{1},x_{1})$ . But, by Cartesianness of the left-hand diagram, the point $\unicode[STIX]{x1D709}_{2}$ is the unique preimage of $(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{1})$ under ${\mathcal{X}}_{2}\rightarrow {\mathcal{X}}_{1}\times {\mathcal{X}}_{1}$ . It follows that the point $x_{2}$ is the unique preimage of $(x_{1},x_{1})$ .
Since $X_{2}\times _{q_{1},X_{1},q_{2}}X_{2}\rightarrow X_{2}$ is an étale equivalence relation, there exist an algebraic space quotient $X$ and induced maps $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ and $X_{1}\rightarrow X$ . Consider
Since ${\mathcal{X}}_{2}\simeq {\mathcal{X}}_{1}\times _{X_{1}}X_{2}$ and $X_{2}\simeq X_{1}\times _{X}X_{1}$ , the left-hand and outer squares above are Cartesian. Since ${\mathcal{X}}_{1}\rightarrow {\mathcal{X}}$ is étale and surjective, it follows that the right-hand square is Cartesian. By descent [Reference AlperAlp13, Proposition 4.7], $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ is a good moduli space.◻
Proof of Theorem 1.2.
After taking a disjoint union of finitely many local quotient presentations, there exists a strongly étale, affine, and surjective morphism $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ , where ${\mathcal{W}}$ admits a good moduli space. The theorem now follows from Proposition 3.1.◻
3.2 Existence via local VGIT
To prove Theorem 1.3, we will need some preliminary lemmas.
Lemma 3.2. Let $G$ be a linearly reductive group scheme over an algebraically closed field $k$ acting on an affine scheme $X$ of finite type over $k$ . Let $\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$ be a non-trivial character. Let $\unicode[STIX]{x1D706}:\mathbb{G}_{m}\rightarrow G$ be a one-parameter subgroup and $x\in X_{\unicode[STIX]{x1D712}}^{-}(k)$ such that $x_{0}=\lim _{t\rightarrow 0}\unicode[STIX]{x1D706}(t)\cdot x\in X^{G}$ is fixed by $G$ . Then $\langle \unicode[STIX]{x1D712},\unicode[STIX]{x1D706}\rangle >0$ .
Proof. As $x\in X_{\unicode[STIX]{x1D712}}^{-}$ , we have $\langle \unicode[STIX]{x1D712},\unicode[STIX]{x1D706}\rangle \geqslant 0$ by the Hilbert–Mumford criterion (cf. [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Proposition 3.5]). Suppose that $\langle \unicode[STIX]{x1D712},\unicode[STIX]{x1D706}\rangle =0$ . Considering the action of $G$ on $X\times \mathbb{A}^{1}$ induced by $\unicode[STIX]{x1D712}^{-1}$ via $g\cdot (x,s)=(g\cdot x,\unicode[STIX]{x1D712}(g)^{-1}s)$ , we obtain
But $X^{G}$ is contained in the unstable locus $X\setminus X_{\unicode[STIX]{x1D712}}^{-}$ since $\unicode[STIX]{x1D712}$ is a non-trivial linearization. It follows that $\overline{G\cdot (x,1)}\cap (X^{G}\times \{0\})\neq \emptyset$ , which contradicts $x\in X_{\unicode[STIX]{x1D712}}^{-}$ (cf. [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Remark 3.2]).◻
Lemma 3.3 [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Lemma 3.9].
Let $G$ be a linearly reductive group scheme over an algebraically closed field $k$ and let $\unicode[STIX]{x1D712}:G\rightarrow \mathbb{G}_{m}$ be a character. Let $h:\operatorname{Spec}A=X\rightarrow Y=\operatorname{Spec}B$ be a $G$ -invariant morphism of affine schemes of finite type over $k$ . Assume that $A=B\otimes _{B^{G}}A^{G}$ . Then $h^{-1}(Y_{\unicode[STIX]{x1D712}}^{+})=X_{\unicode[STIX]{x1D712}}^{+}$ and $h^{-1}(Y_{\unicode[STIX]{x1D712}}^{-})=X_{\unicode[STIX]{x1D712}}^{-}$ .
Remark 3.4. While [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Proposition 3.5 and Lemma 3.9] have an underlying characteristic- $0$ hypothesis, it is immediate that the proofs carry over to any characteristic.
Lemma 3.5. Let ${\mathcal{X}}$ be an algebraic stack of finite type over an algebraically closed field $k$ and let ${\mathcal{L}}$ be a line bundle on ${\mathcal{X}}$ . Let ${\mathcal{X}}^{+},{\mathcal{X}}^{-}\subset {\mathcal{X}}$ be open substacks and let ${\mathcal{Z}}^{+}={\mathcal{X}}\smallsetminus {\mathcal{X}}^{+}$ , ${\mathcal{Z}}^{-}={\mathcal{X}}\smallsetminus {\mathcal{X}}^{-}$ be their reduced complements. Suppose that for all closed points $x\in {\mathcal{X}}$ , there exist a local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ around $x$ and a Cartesian diagram
where ${\mathcal{W}}^{+}={\mathcal{W}}_{{\mathcal{L}}}^{+}$ and ${\mathcal{W}}^{-}={\mathcal{W}}_{{\mathcal{L}}}^{-}$ are the VGIT chambers of ${\mathcal{W}}$ with respect to ${\mathcal{L}}$ . Then:
-
(i) if $z\in {\mathcal{X}}^{+}(k)\cap {\mathcal{X}}^{-}(k)$ , then the closure of $z$ in ${\mathcal{X}}$ is contained in ${\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ ; and
-
(ii) if $z\in {\mathcal{X}}(k)$ is a closed point, then either $z\in {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ or $z\in {\mathcal{Z}}^{+}\cap {\mathcal{Z}}^{-}$ .
Proof. For (i), if the closure of $z$ in ${\mathcal{X}}$ is not contained in ${\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ , then there exists an isotrivial specialization $z{\rightsquigarrow}x$ to a closed point in ${\mathcal{X}}\setminus ({\mathcal{X}}^{+}\cap {\mathcal{X}}^{-})$ . Choose a local quotient presentation $f:{\mathcal{W}}=[W/G_{x}]\rightarrow {\mathcal{X}}$ around $x$ such that (3.1) is Cartesian. Since $f^{-1}(x)\not \subset {\mathcal{W}}^{+}\cap {\mathcal{W}}^{-}$ , the character $\unicode[STIX]{x1D712}={\mathcal{L}}|_{BG_{x}}$ is non-trivial. By the Hilbert–Mumford criterion (see [Reference MumfordMum65, Theorem 2.1]), there exists a one-parameter subgroup $\unicode[STIX]{x1D706}:\mathbb{G}_{m}\rightarrow G_{x}$ such that $\lim _{t\rightarrow 0}\unicode[STIX]{x1D706}(t)\cdot w=w_{0}$ , where $w\in W$ and $w_{0}\in W^{G_{x}}$ are points over $z$ and $x$ , respectively. As $w\in W_{\unicode[STIX]{x1D712}}^{+}\cap W_{\unicode[STIX]{x1D712}}^{-}$ and $w_{0}\in W^{G_{x}}$ , by applying Lemma 3.2 twice with the characters $\unicode[STIX]{x1D712}$ and $\unicode[STIX]{x1D712}^{-1}$ , we see that both $\langle \unicode[STIX]{x1D712},\unicode[STIX]{x1D706}\rangle <0$ and $\langle \unicode[STIX]{x1D712},\unicode[STIX]{x1D706}\rangle >0$ , which is a contradiction.
For (ii), choose a local quotient presentation $f:({\mathcal{W}},w)\rightarrow ({\mathcal{X}},z)$ around $z$ with ${\mathcal{W}}=[W/G_{z}]$ . Let $\unicode[STIX]{x1D712}={\mathcal{L}}|_{BG_{z}}$ be the character of ${\mathcal{L}}$ . Since $w\in W^{G_{z}}$ , the point $w$ is semistable with respect to $\unicode[STIX]{x1D712}$ if and only if $\unicode[STIX]{x1D712}$ is trivial. It follows that $w\in {\mathcal{W}}^{+}\cap {\mathcal{W}}^{-}$ if $\unicode[STIX]{x1D712}$ is trivial and $w\notin {\mathcal{W}}^{+}\cup {\mathcal{W}}^{-}$ otherwise.◻
Proof of Theorem 1.3.
We show that ${\mathcal{X}}$ has a good moduli space by verifying the hypotheses of Theorem 1.2. Let $x_{0}\in {\mathcal{X}}$ be a closed point. By Lemma 3.5(ii), we have either $x_{0}\in {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ or $x_{0}\in {\mathcal{Z}}^{+}\cap {\mathcal{Z}}^{-}$ . Suppose first that $x_{0}\in {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ . Since ${\mathcal{X}}^{+}$ admits a good moduli space, Proposition 2.7(iii) implies that we may choose a local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}^{+}$ around $x_{0}$ which is strongly étale. By applying Lemma 2.8, we may shrink further to assume that $f({\mathcal{W}})\subset {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ . Then Lemma 3.5(i) implies that the composition $f:{\mathcal{W}}\rightarrow {\mathcal{X}}^{+}{\hookrightarrow}{\mathcal{X}}$ is also strongly étale.◻
On the other hand, suppose that $x_{0}\in {\mathcal{Z}}^{+}\cap {\mathcal{Z}}^{-}$ . Choose a local quotient presentation $f:({\mathcal{W}},w_{0})\rightarrow ({\mathcal{X}},x_{0})$ around $x_{0}$ inducing a Cartesian diagram
where ${\mathcal{W}}^{+}={\mathcal{W}}_{{\mathcal{L}}}^{+}$ and ${\mathcal{W}}^{-}={\mathcal{W}}_{{\mathcal{L}}}^{-}$ . We claim that, after shrinking suitably, we may assume that $f$ is strongly étale. In proving this claim, we make implicit repeated use of Lemma 2.8 in conjunction with Lemma 3.3 to argue that if ${\mathcal{W}}^{\prime }\subset {\mathcal{W}}$ is an open substack containing $w_{0}$ , there exists an open substack ${\mathcal{W}}^{\prime \prime }\subset {\mathcal{W}}^{\prime }$ containing $w_{0}$ such that ${\mathcal{W}}^{\prime \prime }\rightarrow {\mathcal{X}}$ is a local quotient presentation around $x_{0}$ inducing a Cartesian diagram as in (3.2).
Using the hypothesis that ${\mathcal{Z}}^{+},{\mathcal{Z}}^{-},$ and ${\mathcal{X}}^{+}$ admit good moduli spaces, we will first show that $f$ may be chosen to satisfy:
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(A) $f|_{f^{-1}({\mathcal{Z}}^{+})}$ , $f|_{f^{-1}({\mathcal{Z}}^{-})}$ are strongly étale; and
-
(B) $f|_{{\mathcal{W}}^{+}}:{\mathcal{W}}^{+}\rightarrow {\mathcal{X}}^{+}$ is strongly étale.
If $f$ satisfies (A) and (B), then $f$ is also strongly étale. Indeed, if $w\in {\mathcal{W}}$ is a closed point, then either $w\in f^{-1}({\mathcal{Z}}^{+})\cup f^{-1}({\mathcal{Z}}^{-})$ or $w\in f^{-1}({\mathcal{X}}^{+})\cap f^{-1}({\mathcal{X}}^{-})$ . In the former case, (A) immediately implies that $f$ is stabilizer preserving at $w$ and that $f(w)$ is closed in ${\mathcal{X}}$ . In the latter case, (B) implies that $f$ is stabilizer preserving at $w$ and that $f(w)$ is closed in ${\mathcal{X}}^{+}$ . Since $f(w)\in {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ , Lemma 3.5(i) implies that $f(w)$ remains closed in ${\mathcal{X}}$ .
It remains to show that $f$ can be chosen to satisfy (A) and (B). For (A), Proposition 2.7(iii) implies the existence of an open substack ${\mathcal{Q}}\subset f^{-1}({\mathcal{Z}}^{+})$ containing $w_{0}$ such that $f|_{{\mathcal{Q}}}$ is strongly étale. After shrinking ${\mathcal{W}}$ suitably, we may assume that ${\mathcal{W}}\cap f^{-1}({\mathcal{Z}}^{+})\subset {\mathcal{Q}}$ . One argues similarly for $f|_{f^{-1}({\mathcal{Z}}^{-})}$ .
For (B), Proposition 2.7(iii) implies that there exists an open substack ${\mathcal{U}}\subset {\mathcal{W}}^{+}$ such that $f|_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow {\mathcal{X}}^{+}$ is strongly étale and, moreover, that ${\mathcal{U}}$ contains all closed points $w\in {\mathcal{W}}^{+}$ such that $f|_{{\mathcal{W}}^{+}}:{\mathcal{W}}^{+}\rightarrow {\mathcal{X}}^{+}$ is strongly étale at $w$ . Let ${\mathcal{V}}={\mathcal{W}}^{+}\setminus {\mathcal{U}}$ and let $\overline{{\mathcal{V}}}$ be the closure of ${\mathcal{V}}$ in ${\mathcal{W}}$ . We claim that $w_{0}\notin \overline{{\mathcal{V}}}$ . Once this is established, we may replace ${\mathcal{W}}$ by an appropriate open substack of ${\mathcal{W}}\setminus \overline{{\mathcal{V}}}$ to obtain a local quotient presentation satisfying (B). Suppose, by way of contradiction, that $w_{0}\in \overline{{\mathcal{V}}}$ . Then there exists a specialization diagram (where $R$ is a discrete valuation ring with fraction field $K$ )
such that $h(0)=w_{0}$ . There exist good moduli spaces ${\mathcal{W}}\rightarrow W$ and ${\mathcal{W}}^{+}\rightarrow W^{+}$ , and the induced morphism $W^{+}\rightarrow W$ is proper. Since the composition ${\mathcal{W}}^{+}\rightarrow W^{+}\rightarrow W$ is universally closed, there exist, after an extension of the fraction field $K$ , a diagram
and a lift $\widetilde{h}:\unicode[STIX]{x1D6E5}\rightarrow {\mathcal{W}}^{+}$ that extends $\unicode[STIX]{x1D6E5}^{\ast }\rightarrow {\mathcal{W}}^{+}$ with $\widetilde{w}=\widetilde{h}(0)\in {\mathcal{W}}^{+}$ closed. There is an isotrivial specialization $\widetilde{w}{\rightsquigarrow}w_{0}$ . It follows from Lemma 3.5(i) that $\widetilde{w}\in f^{-1}({\mathcal{Z}}^{-})$ . By assumption (A), $f|_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow {\mathcal{X}}^{+}$ is strongly étale at $\widetilde{w}$ , so that $\widetilde{w}\in {\mathcal{U}}$ . On the other hand, the generic point of the specialization $\widetilde{h}:\unicode[STIX]{x1D6E5}\rightarrow {\mathcal{W}}^{+}$ lands in ${\mathcal{V}}$ , so that $\widetilde{w}\in {\mathcal{V}}$ , which is a contradiction. Thus, $w_{0}\notin \overline{{\mathcal{V}}}$ , as desired.
We have now shown that ${\mathcal{X}}$ satisfies condition (1) in Theorem 1.2, and it remains to verify condition (2). Let $x\in {\mathcal{X}}(k)$ . If $x\in {\mathcal{Z}}^{+}$ (respectively, $x\in {\mathcal{Z}}^{-}$ ), then $\overline{\{x\}}\subset {\mathcal{Z}}^{+}$ (respectively, $\overline{\{x\}}\subset {\mathcal{Z}}^{-}$ ). Therefore, since ${\mathcal{Z}}^{+}$ (respectively, ${\mathcal{Z}}^{-}$ ) admits a good moduli space, so does $\overline{\{x\}}$ . On the other hand, if $x\in {\mathcal{X}}^{+}\cap {\mathcal{X}}^{-}$ , then Lemma 3.5(i) implies that the closure of $x$ in ${\mathcal{X}}$ is contained in ${\mathcal{X}}^{+}$ . Since ${\mathcal{X}}^{+}$ admits a good moduli space, so does $\overline{\{x\}}$ . Now Theorem 1.2 implies that ${\mathcal{X}}$ admits a good moduli space $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow X$ .
Next, we will apply Proposition 3.1 to show that ${\mathcal{X}}^{-}$ admits a good moduli space. Let $x\in {\mathcal{X}}^{-}$ be a closed point and $x{\rightsquigarrow}x_{0}$ be the isotrivial specialization to the unique closed point $x_{0}\in {\mathcal{X}}$ in its closure. By Proposition 2.7, there exists a strongly étale local quotient presentation $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ around $x_{0}$ inducing a Cartesian diagram as in (1.2). By Lemma 2.9, the base change $f^{-}:{\mathcal{W}}^{-}\rightarrow {\mathcal{X}}^{-}$ is strongly étale. As ${\mathcal{W}}^{-}$ admits a good moduli space, we may shrink ${\mathcal{W}}^{-}$ further so that $f^{-}:{\mathcal{W}}^{-}\rightarrow {\mathcal{X}}^{-}$ is a strongly étale neighborhood of $x$ .
It remains to check that if $x\in {\mathcal{X}}^{-}(k)$ is any point, then its closure $\overline{\{x\}}$ in ${\mathcal{X}}^{-}$ admits a good moduli space. Let $x{\rightsquigarrow}x_{0}$ be the isotrivial specialization to the unique closed point $x_{0}\in {\mathcal{X}}$ in the closure of $x$ . We claim in fact that $\unicode[STIX]{x1D719}^{-1}(\unicode[STIX]{x1D719}(x_{0}))\cap {\mathcal{X}}^{-}$ admits a good moduli space, which in turn clearly implies that $\overline{\{x\}}\subset {\mathcal{X}}^{-}$ does as well. We can choose a local quotient presentation $f:({\mathcal{W}},w_{0})\rightarrow ({\mathcal{X}},x_{0})$ around $x_{0}$ inducing a Cartesian diagram as in (1.2). After shrinking, we may assume by Proposition 2.7(iii) that $f$ is strongly étale and we may also assume that $w_{0}$ is the unique preimage of $x_{0}$ . If we set ${\mathcal{Z}}=\unicode[STIX]{x1D719}^{-1}(\unicode[STIX]{x1D719}(x_{0}))$ , then $f|_{f^{-1}({\mathcal{Z}})}:f^{-1}({\mathcal{Z}})\rightarrow {\mathcal{Z}}$ is in fact an isomorphism as both $f^{-1}({\mathcal{Z}})$ and ${\mathcal{Z}}$ have $\operatorname{Spec}k$ as a good moduli space. As ${\mathcal{W}}^{-}$ admits a good moduli space, so does ${\mathcal{W}}^{-}\cap f^{-1}({\mathcal{Z}})={\mathcal{X}}^{-}\cap {\mathcal{Z}}$ . This establishes that ${\mathcal{X}}^{-}$ admits a good moduli space.
Finally, we argue that $X^{+}\rightarrow X$ and $X^{-}\rightarrow X$ are proper and surjective. By taking a disjoint union of local quotient presentations and applying Proposition 2.7(iii), there exists an affine, strongly étale, and surjective morphism $f:{\mathcal{W}}\rightarrow {\mathcal{X}}$ from an algebraic stack admitting a good moduli space ${\mathcal{W}}\rightarrow W$ such that ${\mathcal{W}}={\mathcal{X}}\times _{X}W$ . Moreover, if we set ${\mathcal{W}}^{+}:=f^{-1}({\mathcal{X}}^{+})$ and ${\mathcal{W}}^{-}:=f^{-1}({\mathcal{X}}^{-})$ , then ${\mathcal{W}}^{+}$ and ${\mathcal{W}}^{-}$ admit good moduli spaces $W^{+}$ and $W^{-}$ such that $W^{-}\rightarrow W$ and $W^{+}\rightarrow W$ are proper and surjective. This gives commutative cubes
As $f|_{{\mathcal{W}}^{+}}:{\mathcal{W}}^{+}\rightarrow {\mathcal{X}}^{+}$ and $f|_{{\mathcal{W}}^{-}}:{\mathcal{W}}^{-}\rightarrow {\mathcal{X}}^{-}$ are strongly étale (Lemma 2.9), the left- and right-hand faces are Cartesian squares. Since the top faces are also Cartesian, we have ${\mathcal{W}}^{+}\simeq {\mathcal{X}}^{+}\times _{X}W$ and ${\mathcal{W}}^{-}\simeq {\mathcal{X}}^{-}\times _{X}W$ . In particular, ${\mathcal{W}}^{+}\rightarrow X^{+}\times _{X}W$ and ${\mathcal{W}}^{-}\rightarrow X^{-}\times _{X}W$ are good moduli spaces. By uniqueness of good moduli spaces, we have $X^{+}\times _{X}W\simeq W^{+}$ and $X^{-}\times _{X}W\simeq W^{-}$ . Since $W^{+}\rightarrow W$ and $W^{-}\rightarrow W$ are proper and surjective, $X^{+}\rightarrow X$ and $X^{-}\rightarrow X$ are proper and surjective by étale descent.
3.3 Existence via finite covers
In proving Proposition 1.4, we will appeal to the following lemma.
Lemma 3.6. Consider a commutative diagram
of algebraic stacks of finite type over an algebraically closed field $k$ of characteristic $0$ , where $X$ is an algebraic space. Suppose that:
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(i) the morphism ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ is finite and surjective;
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(ii) the morphism ${\mathcal{X}}\rightarrow X$ is cohomologically affine; and
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(iii) the algebraic stack ${\mathcal{Y}}$ is a global quotient stack.
Then ${\mathcal{Y}}\rightarrow X$ is cohomologically affine.
Proof. We may write ${\mathcal{Y}}=[V/\text{GL}_{n}]$ , where $V$ is an algebraic space with an action of $\operatorname{GL}_{n}$ . Since ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ is affine, ${\mathcal{X}}$ is the quotient stack ${\mathcal{X}}=[U/\text{GL}_{n}]$ , where $U={\mathcal{X}}\times _{{\mathcal{Y}}}V$ . Since $U\rightarrow {\mathcal{X}}$ is affine and ${\mathcal{X}}\rightarrow X$ is cohomologically affine, the morphism $U\rightarrow X$ is affine by Serre’s criterion. The morphism $U\rightarrow V$ is finite and surjective and therefore, by Chevalley’s theorem, we can conclude that $V\rightarrow X$ is affine. Therefore, ${\mathcal{Y}}\rightarrow X$ is cohomologically affine.◻
Proof of Proposition 1.4.
Let ${\mathcal{Z}}$ be the scheme-theoretic image of ${\mathcal{X}}\rightarrow X\times {\mathcal{Y}}$ . Since ${\mathcal{X}}\rightarrow {\mathcal{Y}}$ is finite and $X$ is separated, ${\mathcal{X}}\rightarrow {\mathcal{Z}}$ is finite. Since ${\mathcal{Z}}$ is a global quotient stack (as ${\mathcal{Y}}$ is), we may apply Lemma 3.6 to conclude that the projection ${\mathcal{Z}}\rightarrow X$ is cohomologically affine, which implies that ${\mathcal{Z}}$ admits a separated good moduli space. The composition ${\mathcal{Z}}{\hookrightarrow}X\times {\mathcal{Y}}\rightarrow {\mathcal{Y}}$ is finite, surjective, and stabilizer preserving at closed points. Therefore, by replacing ${\mathcal{X}}$ with ${\mathcal{Z}}$ , to prove the proposition, we may assume that $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ is stabilizer preserving at closed points.◻
We will now show that the hypotheses of Theorem 1.2 are satisfied. Let $y_{0}\in {\mathcal{Y}}$ be a closed point and $g:({\mathcal{Y}}^{\prime },y_{0}^{\prime })\rightarrow ({\mathcal{Y}},y_{0})$ be a local quotient presentation around $y_{0}$ . Consider the Cartesian diagram
We first note that $g^{\prime }$ is strongly étale at each point $x^{\prime }\in {f^{\prime }}^{-1}(y_{0}^{\prime })$ . Indeed, $g^{\prime }$ is stabilizer preserving at $x^{\prime }$ as $g$ is stabilizer preserving at $y_{0}^{\prime }$ , and $g^{\prime }(x^{\prime })\in {\mathcal{X}}$ is closed as $f(g^{\prime }(x^{\prime }))\in {\mathcal{Y}}$ is closed. By Proposition 2.7, there exists an open substack ${\mathcal{U}}^{\prime }\subset {\mathcal{X}}^{\prime }$ containing the fiber over $y_{0}^{\prime }$ such that $g^{\prime }|_{{\mathcal{U}}^{\prime }}$ is strongly étale. Therefore, $y_{0}^{\prime }\notin {\mathcal{Z}}=f^{\prime }({\mathcal{X}}^{\prime }\setminus {\mathcal{U}}^{\prime })$ and $g|_{{\mathcal{Y}}^{\prime }\setminus {\mathcal{Z}}}$ is strongly étale. By shrinking further using Lemma 2.8, we obtain a local quotient presentation $g:{\mathcal{Y}}^{\prime }\rightarrow {\mathcal{Y}}$ around $y_{0}$ which is strongly étale.
Finally, let $y\in {\mathcal{Y}}(k)$ and $x\in {\mathcal{X}}(k)$ be any preimage. Set ${\mathcal{X}}_{0}=\overline{\{x\}}\subset {\mathcal{X}}$ and ${\mathcal{Y}}_{0}=\overline{\{y\}}\subset {\mathcal{Y}}$ . As ${\mathcal{X}}_{0}\rightarrow {\mathcal{Y}}_{0}$ is finite and surjective, ${\mathcal{X}}_{0}\rightarrow \operatorname{Spec}k$ is a good moduli space, and ${\mathcal{Y}}_{0}$ is a global quotient stack, we may conclude using Lemma 3.6 that ${\mathcal{Y}}_{0}$ admits a good moduli space. Therefore, we may apply Theorem 1.2 to establish the proposition.
Remark.
The hypothesis that $X$ is separated in Proposition 1.4 is necessary. For example, let $X$ be the affine line with $0$ doubled and let $\mathbb{Z}_{2}$ act on $X$ by swapping the points at $0$ and fixing all other points. Then $X\rightarrow [X/\mathbb{Z}_{2}]$ satisfies the hypotheses but $[X/\mathbb{Z}_{2}]$ does not admit a good moduli space.
4 Application to $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$
In this section, we apply Theorem 1.3 to prove that the algebraic stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ admit good moduli spaces (Theorem 1.1). We already know that the inclusions
arise from local VGIT with respect to $\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D713}$ [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Theorem 3.17]. Thus, it only remains to show that for each critical value $\unicode[STIX]{x1D6FC}_{c}\in \{9/11,7/10,2/3\}$ , the closed substacks
admit good moduli spaces. We will prove this statement by induction on $g$ . As with the boundary strata of $\overline{{\mathcal{M}}}_{g,n}$ , $\overline{{\mathcal{H}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ can be described (up to a finite cover) as a product of moduli spaces of $\unicode[STIX]{x1D6FC}_{c}$ -stable curves of lower genus. Likewise, $\overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ can be described (up to a finite cover) as stacky projective bundles over moduli spaces of $\unicode[STIX]{x1D6FC}_{c}$ -stable curves of lower genus. We use induction to deduce that these products and projective bundles admit good moduli spaces, and then apply Proposition 1.4 to conclude that $\overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ and $\overline{{\mathcal{H}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ admit good moduli spaces.
4.1 Properties of $\unicode[STIX]{x1D6FC}$ -stability
We will make repeated use of the following elementary properties of $\unicode[STIX]{x1D6FC}$ -stability.
Lemma 4.1 [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Lemma 2.17].
(i) If $(\widetilde{C}_{1},\{p_{i}\}_{i=1}^{n},q_{1})$ and $(\widetilde{C}_{2},\{p_{i}\}_{i=1}^{n},q_{2})$ are $\unicode[STIX]{x1D6FC}$ -stable curves, then
is $\unicode[STIX]{x1D6FC}$ -stable.
(ii) If $(\widetilde{C},\{p_{i}\}_{i=1}^{n},q_{1},q_{2})$ is an $\unicode[STIX]{x1D6FC}$ -stable curve, then
is $\unicode[STIX]{x1D6FC}$ -stable provided one of the following conditions hold:
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– $q_{1}$ and $q_{2}$ lie on disjoint irreducible components of $\widetilde{C}$ ;
-
– $q_{1}$ and $q_{2}$ lie on distinct irreducible components of $\widetilde{C}$ , and at least one of these components is not a smooth rational curve.
Lemma 4.2 [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Lemma 2.19].
Let $(C,\{p_{i}\}_{i=1}^{n})$ be an $n$ -pointed curve with $\unicode[STIX]{x1D714}_{C}(\sum _{i}p_{i})$ ample, and suppose that $q\in C$ is an $\unicode[STIX]{x1D6FC}_{c}$ -critical singularity. Then the stable pointed normalization of $(C,\{p_{i}\}_{i=1}^{n})$ at $q\in C$ is $\unicode[STIX]{x1D6FC}_{c}$ -stable if and only if $(C,\{p_{i}\}_{i=1}^{n})$ is $\unicode[STIX]{x1D6FC}_{c}$ -stable.
4.2 Existence for $\overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$
In this section, we use induction on $g$ to prove that $\overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ admits a good moduli space. The base case is handled by the following lemma.
Lemma 4.3. We have
In particular, the algebraic stacks $\overline{{\mathcal{S}}}_{1,1}(9/11)$ , $\overline{{\mathcal{S}}}_{1,2}(7/10)$ , and $\overline{{\mathcal{S}}}_{2,1}(2/3)$ admit good moduli spaces.
Proof. The algebraic stacks $\overline{{\mathcal{S}}}_{1,1}(9/11)$ and $\overline{{\mathcal{S}}}_{1,2}(7/10)$ each contain a unique