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].◻

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 $U_{1}\subset X\setminus Z$ .◻

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.◻

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.◻

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

$$\begin{eqnarray}\lim _{t\rightarrow 0}\unicode[STIX]{x1D706}(t)\cdot (x,1)=(x_{0},1)\in X^{G}\times \mathbb{A}^{1}.\end{eqnarray}$$

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]).◻

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.◻

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.◻

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.

Proof. The algebraic stacks $\overline{{\mathcal{S}}}_{1,1}(9/11)$ and $\overline{{\mathcal{S}}}_{1,2}(7/10)$ each contain a unique $\mathbb{C}$ -point, namely the $9/11$ -atom and the $7/10$ -atom, and each of these curves has a $\mathbb{G}_{m}$ -automorphism group. The stack $\overline{{\mathcal{S}}}_{2,1}(2/3)$ contains two isomorphism classes of curves, namely the $2/3$ -atom, and the rational ramphoid cuspidal curve with non-trivial crimping. We construct this stack explicitly as follows: start with the constant family $(\mathbb{P}^{1}\times \mathbb{A}^{1},\infty \times \mathbb{A}^{1})$ , let $c$ be a coordinate on $\mathbb{A}^{1}$ , and $t$ a coordinate on $\mathbb{P}^{1}-\infty$ . Now let $\mathbb{P}^{1}\times \mathbb{A}^{1}\rightarrow {\mathcal{C}}$ be the map defined by the inclusion of algebras $\mathbb{C}[t^{2}+ct^{3},t^{5}]\subset \mathbb{C}[c,t]$ on the complement of the infinity section, and defined as an isomorphism on the complement of the zero section. Then $({\mathcal{C}}\rightarrow \mathbb{A}^{1},\infty \times \mathbb{A}^{1})$ is a family of rational ramphoid cuspidal curves whose fiber over $0\in \mathbb{A}^{1}$ is a $2/3$ -atom. Furthermore, $\mathbb{G}_{m}$ acts on the base and total space of this family by $t\rightarrow \unicode[STIX]{x1D706}^{-1}t,c\rightarrow \unicode[STIX]{x1D706}c$ , since the subalgebra $\mathbb{C}[t^{2}+ct^{3},t^{5}]\subset \mathbb{C}[c,t]$ is invariant under this action. Thus, the family descends to $[\mathbb{A}^{1}/\mathbb{G}_{m}]$ and there is an induced map $[\mathbb{A}^{1}/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{2,1}(2/3)$ . This map is a locally closed immersion by [Reference van der WyckVdW10, Theorem 1.109], and the image is precisely $\overline{{\mathcal{S}}}_{2,1}(2/3)$ . Thus, $\overline{{\mathcal{S}}}_{2,1}(2/3)\simeq [\mathbb{A}^{1}/\mathbb{G}_{m}]$ , as desired.◻

Remark.

Suppose that $C^{\prime }$ is a cuspidal sprouting, one-sided tacnodal sprouting, or ramphoid cuspidal sprouting of $(C,p_{1})$ (respectively, tacnodal sprouting of $(C,p_{1},p_{2})$ ) with $\unicode[STIX]{x1D6FC}_{c}$ -critical singularity $q\in C^{\prime }$ . Then $(C,p_{1})$ (respectively, $(C,p_{1},p_{2})$ ) is the stable pointed normalization of $C^{\prime }$ along $q$ . By Lemma 4.2, $C^{\prime }$ is $\unicode[STIX]{x1D6FC}_{c}$ -stable if and only if $(C,p_{1})$ (respectively, $(C,p_{1},p_{2})$ ) is $\unicode[STIX]{x1D6FC}_{c}$ -stable.

Proof. If $(C,\{p_{i}\}_{i=1}^{n})\in \overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ , then $(C,\{p_{i}\}_{i=1}^{n})$ contains an $\unicode[STIX]{x1D6FC}_{c}$ -critical singularity $q\in C$ . The stable pointed normalization of $(C,\{p_{i}\}_{i=1}^{n})$ along $q$ is well defined by our hypothesis on $(g,n)$ , and is $\unicode[STIX]{x1D6FC}_{c}$ -stable by Lemma 4.2.◻

Proof. It is clear that $F_{k}$ is representable. The fact that $F_{k}$ is quasi-finite follows from the observations that a curve $(C,\{p_{i}\}_{i=1}^{n})$ in ${\mathcal{U}}_{g,n}(A_{k})$ has only a finite number of $A_{k}$ -singularities and that for a $\mathbb{C}$ -point $x\in \operatorname{Sprout}_{g,n}(A_{k})$ , the induced map $\operatorname{Aut}_{\operatorname{Sprout}_{g,n}(A_{k})}(x)\rightarrow \operatorname{Aut}_{{\mathcal{U}}_{g,n}(A_{k})}(F_{k}(x))$ on automorphism groups has finite cokernel. To show that $F_{k}$ is finite, it now suffices to verify the valuative criterion for properness: let $\unicode[STIX]{x1D6E5}$ be the spectrum of a discrete valuation ring, let $\unicode[STIX]{x1D6E5}^{\ast }$ denote the spectrum of its fraction field, and suppose that we are given a diagram

This corresponds to a diagram of families,

such that ${\mathcal{C}}_{\unicode[STIX]{x1D6E5}^{\ast }}$ has an $A_{k}$ -singularity along $\unicode[STIX]{x1D70E}_{n+1}$ . Since ${\mathcal{C}}\rightarrow \unicode[STIX]{x1D6E5}$ is proper, $\unicode[STIX]{x1D70E}_{n+1}$ extends uniquely to a section of $\unicode[STIX]{x1D70B}$ and, since the limit of an $A_{k}$ -singularity in ${\mathcal{U}}_{g,n}(A_{k})$ is necessarily an $A_{k}$ -singularity, ${\mathcal{C}}$ has an $A_{k}$ -singularity along $\unicode[STIX]{x1D70E}_{n+1}$ . This induces a unique lift $\unicode[STIX]{x1D6E5}\rightarrow \operatorname{Sprout}_{g,n}(A_{k})$ ; cf. [Reference van der WyckVdW10, Theorem 1.109].◻

Proof. We prove the hardest case (v), and leave the others as an exercise to the reader. To construct a map $\unicode[STIX]{x1D6FE}:[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \operatorname{Sprout}_{g,n}(A_{4})$ , we start with a family $(\unicode[STIX]{x1D70B}:{\mathcal{C}}\rightarrow X,\{\unicode[STIX]{x1D70E}_{i}\}_{i=1}^{n+1})$ in ${\mathcal{U}}_{g-2,n+1}(A_{4})$ , and construct a family of ramphoid cuspidal sproutings over $[V({\mathcal{E}}_{X})/\mathbb{G}_{m}]$ , where

$$\begin{eqnarray}{\mathcal{E}}_{X}:=\unicode[STIX]{x1D70B}_{\ast }({\mathcal{O}}_{{\mathcal{C}}}(-2\unicode[STIX]{x1D70E}_{n+1})/{\mathcal{O}}_{{\mathcal{C}}}(-4\unicode[STIX]{x1D70E}_{n+1})).\end{eqnarray}$$

Let $V:=V({\mathcal{E}}_{X})$ , $p:V\rightarrow X$ the natural projection, and $({\mathcal{C}}_{V}\rightarrow V,\unicode[STIX]{x1D70E}_{V})$ the family obtained from $({\mathcal{C}}\rightarrow X,\unicode[STIX]{x1D70E}_{n+1})$ by base change along $p$ . As the construction is local around $\unicode[STIX]{x1D70E}_{n+1}$ , we will not keep track of $\{\unicode[STIX]{x1D70E}_{i}\}_{i=1}^{n}$ for the remainder of the argument. If we set ${\mathcal{E}}_{V}=p^{\ast }{\mathcal{E}}_{X}$ , there existsa tautological section $e:{\mathcal{O}}_{V}\rightarrow {\mathcal{E}}_{V}$ . Let $Z\subset V$ denote the divisor along which the composition

$$\begin{eqnarray}{\mathcal{O}}_{V}\rightarrow {\mathcal{E}}_{V}\rightarrow (\unicode[STIX]{x1D70B}_{V})_{\ast }({\mathcal{O}}_{{\mathcal{C}}_{V}}(-2\unicode[STIX]{x1D70E}_{V})/{\mathcal{O}}_{{\mathcal{C}}_{V}}(-3\unicode[STIX]{x1D70E}_{V}))\end{eqnarray}$$

vanishes, and let $\unicode[STIX]{x1D719}:\widetilde{{\mathcal{C}}}\rightarrow {\mathcal{C}}_{V}$ be the blow-up of ${\mathcal{C}}_{V}$ along $\unicode[STIX]{x1D70E}_{V}(Z)$ . Since $\unicode[STIX]{x1D70E}_{V}(Z)\subset {\mathcal{C}}_{V}$ is a regular subscheme of codimension 2, the exceptional divisor $E$ of the blow-up is a $\mathbb{P}^{1}$ -bundle over $\unicode[STIX]{x1D70E}_{V}(Z)$ . In other words, for all $z\in Z$ , we have

$$\begin{eqnarray}\widetilde{{\mathcal{C}}}_{z}={\mathcal{C}}_{z}\cup E_{z}={\mathcal{C}}_{z}\cup \mathbb{P}^{1}.\end{eqnarray}$$

Let $\tilde{\unicode[STIX]{x1D70E}}$ be the strict transform of $\unicode[STIX]{x1D70E}_{V}$ on $\widetilde{{\mathcal{C}}}$ , and observe that $\tilde{\unicode[STIX]{x1D70E}}$ passes through a smooth point of the $\mathbb{P}^{1}$ component in every fiber over $Z$ . We will construct a map $\widetilde{{\mathcal{C}}}\rightarrow {\mathcal{C}}^{\prime }$ which crimps $\tilde{\unicode[STIX]{x1D70E}}$ to a ramphoid cusp, and ${\mathcal{C}}^{\prime }\rightarrow X$ will be the desired family of ramphoid cuspidal sproutings.

Setting $\widetilde{\unicode[STIX]{x1D70B}}:\widetilde{{\mathcal{C}}}\rightarrow {\mathcal{C}}_{V}\rightarrow V$ and

$$\begin{eqnarray}\widetilde{{\mathcal{E}}}=\widetilde{\unicode[STIX]{x1D70B}}_{\ast }({\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-2\tilde{\unicode[STIX]{x1D70E}})/{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-4\tilde{\unicode[STIX]{x1D70E}})),\end{eqnarray}$$

we claim that $e$ induces a section $\widetilde{e}:{\mathcal{O}}_{V}\rightarrow \widetilde{{\mathcal{E}}}$ with the property that the composition

$$\begin{eqnarray}{\mathcal{O}}_{V}\rightarrow \widetilde{{\mathcal{E}}}\rightarrow \widetilde{\unicode[STIX]{x1D70B}}_{\ast }({\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-2\tilde{\unicode[STIX]{x1D70E}})/{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-3\tilde{\unicode[STIX]{x1D70E}}))\end{eqnarray}$$

is never zero. To see this, let $U=\operatorname{Spec}R\subset X$ be an open affine subscheme along which ${\mathcal{E}}$ is trivial, and choose local coordinates $a,b$ on $p^{-1}(U)=\operatorname{Spec}R[a,b]$ such that the tautological section $e$ is given by $at^{2}+bt^{3}$ , where $t$ is a local equation for $\unicode[STIX]{x1D70E}_{V}$ on ${\mathcal{C}}_{V}$ . In these coordinates, $\unicode[STIX]{x1D719}$ is the blow-up along $a=t=0$ . Let $\tilde{a},\tilde{t}$ be homogeneous coordinates for the blow-up and note that on the chart $\tilde{a}\neq 0$ , $t^{\prime }:=\tilde{t}/\tilde{a}$ gives a local equation for $\tilde{\unicode[STIX]{x1D70E}}_{V}$ . In these coordinates, $\unicode[STIX]{x1D719}$ is given by

$$\begin{eqnarray}(a,b,t)\rightarrow (a,b,at^{\prime }).\end{eqnarray}$$

The section $at^{2}+bt^{3}$ pulls back to $a^{3}({t^{\prime }}^{2}+b{t^{\prime }}^{3})$ , and ${t^{\prime }}^{2}+b{t^{\prime }}^{3}$ defines a section $\widetilde{e}$ of $\widetilde{{\mathcal{E}}}$ over $p^{-1}(U)$ with the stated property.

We will use $\widetilde{e}$ to construct a map $\unicode[STIX]{x1D713}:\widetilde{{\mathcal{C}}}\rightarrow {\mathcal{C}}^{\prime }$ such that ${\mathcal{C}}^{\prime }$ has a ramphoid cusp along $\unicode[STIX]{x1D713}\circ \tilde{\unicode[STIX]{x1D70E}}$ . It is sufficient to define $\unicode[STIX]{x1D713}$ locally around $\tilde{\unicode[STIX]{x1D70E}}$ , so we may assume that $\widetilde{\unicode[STIX]{x1D70B}}$ is affine, i.e. $\widetilde{{\mathcal{C}}}:=\operatorname{Spec}_{V}\widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}.$ We specify a sheaf of ${\mathcal{O}}_{V}$ -subalgebras of $\widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}$ as follows: consider the exact sequence

$$\begin{eqnarray}0\rightarrow \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}_{V}}(-4\tilde{\unicode[STIX]{x1D70E}})\rightarrow \tilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}_{V}}(-2\tilde{\unicode[STIX]{x1D70E}})\rightarrow \widetilde{{\mathcal{E}}}\rightarrow 0\end{eqnarray}$$

and let ${\mathcal{F}}\subset \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}$ be the sheaf of ${\mathcal{O}}_{V}$ -subalgebras generated by any inverse image of $\widetilde{e}$ and all functions in $\widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-4\tilde{\unicode[STIX]{x1D70E}})$ . We let $\unicode[STIX]{x1D713}:\operatorname{Spec}_{V}\widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}\rightarrow {\mathcal{C}}^{\prime }:=\operatorname{Spec}_{V}{\mathcal{F}}$ be the map corresponding to the inclusion ${\mathcal{F}}\subset \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}$ . By construction, the inclusion of the complete local rings $\widehat{{\mathcal{O}}}_{C_{v}^{\prime },(\unicode[STIX]{x1D713}\circ \tilde{\unicode[STIX]{x1D70E}})(v)}\subset \widehat{{\mathcal{O}}}_{\widetilde{C}_{v},\tilde{\unicode[STIX]{x1D70E}}(v)}\simeq \mathbb{C}[[t]]$ is of the form $\mathbb{C}[[t^{2}+bt^{3},t^{5}]]\subset \mathbb{C}[[t]]$ for every $v\in V$ . Observe that $\mathbb{C}[[t^{2}+bt^{3},t^{5}]]$ is isomorphic to $\mathbb{C}[[x,y]]/(y^{2}-x^{5})$ as a $\mathbb{C}$ -algebra.

Finally, we claim that ${\mathcal{C}}^{\prime }\rightarrow V$ descends to a family of ramphoid cuspidal sproutings over the quotient stack $[V/\mathbb{G}_{m}]$ . It suffices to show that the subsheaf ${\mathcal{F}}\subset \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}$ is invariant under the natural action of $\mathbb{G}_{m}$ on $V$ . Using the same local coordinates introduced above, the sheaf ${\mathcal{F}}$ is given over the open set $\operatorname{Spec}R[a,b]$ by the $R[a,b]$ -algebra generated by ${t^{\prime }}^{2}+b{t^{\prime }}^{3}$ and ${t^{\prime }}^{5}$ , where $t^{\prime }$ is a local equation for $\tilde{\unicode[STIX]{x1D70E}}$ on $\widetilde{{\mathcal{C}}}$ . To see that this algebra is $\mathbb{G}_{m}$ -invariant, note that the $\mathbb{G}_{m}$ -action on $V=\operatorname{Spec}R[a,b]$ (acting with weight 1 on $a$ and $b$ ) extends canonically to a $\mathbb{G}_{m}$ -action on the blow-up, where $\mathbb{G}_{m}$ acts on $\tilde{a}$ and $\tilde{t}$ with weights $1$ and $0$ , respectively. Thus, $\mathbb{G}_{m}$ acts on $t^{\prime }=\tilde{t}/\tilde{a}$ with weight $-1$ , so that ${t^{\prime }}^{2}+b{t^{\prime }}^{3}$ is a semi-invariant. It follows that the algebra generated by ${t^{\prime }}^{2}+b{t^{\prime }}^{3}$ and ${t^{\prime }}^{5}$ is $\mathbb{G}_{m}$ -invariant, as desired. Thus, we obtain a family $({\mathcal{C}}^{\prime }\rightarrow [V/\mathbb{G}_{m}],\unicode[STIX]{x1D713}\circ \tilde{\unicode[STIX]{x1D70E}})$ in $\operatorname{Sprout}_{g,n}(A_{4})$ , as desired.

To define an inverse map $\unicode[STIX]{x1D6FE}^{-1}:\operatorname{Sprout}_{g,n}(A_{4})\rightarrow [V/\mathbb{G}_{m}]$ , we start with a family $({\mathcal{C}}\rightarrow X,\unicode[STIX]{x1D70E})$ in ${\mathcal{U}}_{g,n}(A_{4})$ such that ${\mathcal{C}}$ has an $A_{4}$ -singularity along $\unicode[STIX]{x1D70E}$ . We must construct a map $X\rightarrow [V({\mathcal{E}})/\mathbb{G}_{m}]$ . By taking the stable pointed normalization of ${\mathcal{C}}$ along $\unicode[STIX]{x1D70E}$ , we obtain a diagram

satisfying:

1. (i) $(\widetilde{{\mathcal{C}}}\rightarrow X,\tilde{\unicode[STIX]{x1D70E}})$ is a family of $(n+1)$ -pointed curves;

2. (ii) $\unicode[STIX]{x1D713}$ is the pointed normalization of ${\mathcal{C}}$ along $\unicode[STIX]{x1D70E}$ , i.e. $\unicode[STIX]{x1D713}$ is finite and restricts to an isomorphism on the open set $\widetilde{{\mathcal{C}}}-\tilde{\unicode[STIX]{x1D70E}}$ ; and

3. (iii) $\unicode[STIX]{x1D719}$ is the stabilization of $(\widetilde{{\mathcal{C}}},\tilde{\unicode[STIX]{x1D70E}})$ , i.e. $\unicode[STIX]{x1D719}$ is the morphism associated to a high multiple of the relatively nef line bundle $\unicode[STIX]{x1D714}_{\widetilde{{\mathcal{C}}}/X}(\tilde{\unicode[STIX]{x1D70E}})$ .

The family $({\mathcal{C}}^{s}\rightarrow X,\unicode[STIX]{x1D70E}_{i}^{s})$ induces a map $X\rightarrow {\mathcal{U}}_{g-2,n+1}(A_{4})$ , and we must show that this lifts to define a map $X\rightarrow [V({\mathcal{E}})/\mathbb{G}_{m}]$ . To see this, let ${\mathcal{F}}$ be the coherent sheaf defined by the following exact sequence:

$$\begin{eqnarray}0\rightarrow \unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{C}}}\cap \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-4\tilde{\unicode[STIX]{x1D70E}})\subset \unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{C}}}\cap \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-2\tilde{\unicode[STIX]{x1D70E}})\rightarrow {\mathcal{F}}\rightarrow 0.\end{eqnarray}$$

The condition that ${\mathcal{C}}$ has a ramphoid cusp along $\unicode[STIX]{x1D713}\circ \tilde{\unicode[STIX]{x1D70E}}$ implies that ${\mathcal{F}}\subset \widetilde{\unicode[STIX]{x1D70B}}_{\ast }{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-2\unicode[STIX]{x1D70E})/{\mathcal{O}}_{\widetilde{{\mathcal{C}}}}(-4\unicode[STIX]{x1D70E})$ is a rank-one sub-bundle. In particular, ${\mathcal{F}}$ induces a sub-bundle of $\unicode[STIX]{x1D70B}_{\ast }^{s}{\mathcal{O}}_{{\mathcal{C}}^{s}}(-2\unicode[STIX]{x1D70E}^{s})/{\mathcal{O}}_{{\mathcal{C}}^{s}}(-4\unicode[STIX]{x1D70E}^{s})$ over the locus of fibers on which $\unicode[STIX]{x1D719}$ is an isomorphism. A local computation, similar to the one performed in the definition of $\unicode[STIX]{x1D6FE}$ , shows that ${\mathcal{F}}$ extends to a subsheaf of $\unicode[STIX]{x1D70B}_{\ast }^{s}{\mathcal{O}}_{{\mathcal{C}}^{s}}(-2\unicode[STIX]{x1D70E}^{s})/{\mathcal{O}}_{{\mathcal{C}}^{s}}(-4\unicode[STIX]{x1D70E}^{s})$ over all of $X$ (though not a sub-bundle; the morphism on fibers is zero precisely where $\unicode[STIX]{x1D719}$ fails to be an isomorphism). The subsheaf ${\mathcal{F}}\subset {\mathcal{E}}$ induces the desired morphism $X\rightarrow [V({\mathcal{E}})/\mathbb{G}_{m}]$ .◻

Proof. Let $\unicode[STIX]{x1D6FC}_{c}=9/11$ . By Lemma 4.3, we may assume that $(g,n)\neq (1,1)$ . By Proposition 4.9(i), there is a locally free sheaf ${\mathcal{E}}$ on $\overline{{\mathcal{M}}}_{g-1,n+1}(9/11)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of cuspidal sproutings of curves in $\overline{{\mathcal{M}}}_{g-1,n+1}(9/11)$ . By Lemma 4.2, the fibers of this family are $9/11$ -stable, so there is an induced map

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(9/11).\end{eqnarray}$$

By Lemma 4.5, $\unicode[STIX]{x1D6F9}$ maps surjectively onto $\overline{{\mathcal{S}}}_{g,n}(9/11)$ . Furthermore, $\unicode[STIX]{x1D6F9}$ is finite by Proposition 4.7. By hypothesis, $\overline{{\mathcal{M}}}_{g-1,n+1}(9/11)$ and therefore $[V({\mathcal{E}})/\mathbb{G}_{m}]$ admits a proper good moduli space. Thus, $\overline{{\mathcal{S}}}_{g,n}(9/11)$ admits a proper good moduli space by Proposition 1.4.

Let $\unicode[STIX]{x1D6FC}_{c}=7/10$ . By Lemma 4.3, we may assume that $(g,n)\neq (1,2)$ . If $g\geqslant 2$ , Proposition 4.9(ii) provides a locally free sheaf ${\mathcal{E}}$ on $\overline{{\mathcal{M}}}_{g-2,n+2}(7/10)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of tacnodal sproutings of curves in $\overline{{\mathcal{M}}}_{g-2,n+2}(7/10)$ , and there is an induced map $[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(7/10).$ Similarly, for every pair of integers $(i,m)$ such that $\overline{{\mathcal{M}}}_{g-i-1,n-m+1}(7/10)\times \overline{{\mathcal{M}}}_{i,m+1}(7/10)$ is defined, by Proposition 4.9(iii), there is a locally free sheaf ${\mathcal{E}}$ on $\overline{{\mathcal{M}}}_{g-i-1,n-m+1}(7/10)\times \overline{{\mathcal{M}}}_{i,m+1}(7/10)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the universal family of tacnodal sproutings. By Lemma 4.2, there are induced maps $[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(7/10).$ Finally, Proposition 4.9(iv) provides a locally free sheaf on $\overline{{\mathcal{M}}}_{g-1,n}(7/10)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of one-sided tacnodal sproutings of curves in $\overline{{\mathcal{M}}}_{g-1,n}(7/10)$ . By Lemma 4.2, there is an induced map $[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(7/10).$ The union of the maps $[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(7/10)$ covers $\overline{{\mathcal{S}}}_{g,n}(7/10)$ by Lemma 4.5. Furthermore, each map is finite by Proposition 4.7. By hypothesis, each of the stacky projective bundles $[V({\mathcal{E}})/\mathbb{G}_{m}]$ admits a proper good moduli space, and therefore so does $\overline{{\mathcal{S}}}_{g,n}(7/10)$ by Proposition 1.4.

Let $\unicode[STIX]{x1D6FC}_{c}=2/3$ . By Lemma 4.3, we may assume that $(g,n)\neq (2,1)$ . By Proposition 4.9(v), there is a locally free sheaf ${\mathcal{E}}$ on $\overline{{\mathcal{M}}}_{g-2,n+1}(2/3)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of ramphoid cuspidal sproutings of curves in $\overline{{\mathcal{M}}}_{g-2,n+1}(2/3)$ . By Lemma 4.2, there is an induced map $\unicode[STIX]{x1D6F9}:[V({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{g,n}(2/3)$ which maps surjectively onto $\overline{{\mathcal{S}}}_{g,n}(2/3)$ by Lemma 4.5. Furthermore, $\unicode[STIX]{x1D6F9}$ is finite by Proposition 4.7. Thus, $\overline{{\mathcal{S}}}_{g,n}(2/3)$ admits a proper good moduli space by Proposition 1.4.◻

Proof. We describe the case of $\overline{{\mathcal{H}}}_{2,1}(2/3)$ , as the other two are essentially identical. Consider the family of Weierstrass tails over $\mathbb{A}^{4}$ given by

$$\begin{eqnarray}y^{2}=x^{5}z+a_{3}x^{3}z^{3}+a_{2}x^{2}z^{4}+a_{1}xz^{5}+a_{0}z^{6},\end{eqnarray}$$

where the Weierstrass section is given by $[1,0,0]$ . Since $\mathbb{G}_{m}$ acts on the base and total space of this family by

$$\begin{eqnarray}x\rightarrow \unicode[STIX]{x1D706}^{2}x,\quad y\rightarrow \unicode[STIX]{x1D706}^{5}y,\quad a_{i}\rightarrow \unicode[STIX]{x1D706}^{10-2i}a_{i},\end{eqnarray}$$

the family descends to $[\mathbb{A}^{4}/\mathbb{G}_{m}]$ . One checks that the induced map $[\mathbb{A}^{4}/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{H}}}_{2,1}(2/3)$ is an isomorphism.◻

Proof. For elliptic chains, Lemma 4.11 handles the case $r=1$ as ${\mathcal{E}}{\mathcal{C}}_{1}=\overline{{\mathcal{H}}}_{1,2}(7/10)$ . By induction on $r$ , we may assume that ${\mathcal{E}}{\mathcal{C}}_{r-1}$ admits a good moduli space. By Proposition 4.9(iii), there is a locally free sheaf ${\mathcal{E}}$ on ${\mathcal{E}}{\mathcal{C}}_{r-1}\times \overline{{\mathcal{H}}}_{1,2}(7/10)$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of tacnodal sproutings over ${\mathcal{E}}{\mathcal{C}}_{r-1}\times \overline{{\mathcal{H}}}_{1,2}(7/10)$ . By Lemma 4.2, there is an induced morphism $\unicode[STIX]{x1D6F9}:[{\mathcal{V}}({\mathcal{E}})/\mathbb{G}_{m}]\rightarrow \overline{{\mathcal{M}}}_{2r-1,2}(7/10).$ The image of $\unicode[STIX]{x1D6F9}$ is ${\mathcal{E}}{\mathcal{C}}_{r}$ , and $\unicode[STIX]{x1D6F9}$ is finite by Proposition 4.7. Since ${\mathcal{E}}{\mathcal{C}}_{r-1}\times \overline{{\mathcal{H}}}_{1,2}(7/10)$ admits a good moduli space, Proposition 1.4 implies that ${\mathcal{E}}{\mathcal{C}}_{r}$ admits a good moduli space.

For Weierstrass chains, Lemma 4.11 again handles the case $r=1$ as ${\mathcal{W}}{\mathcal{C}}_{1}=\overline{{\mathcal{H}}}_{2,1}(2/3)$ . By induction, we may assume that ${\mathcal{W}}{\mathcal{C}}_{r-1}$ admits a good moduli space. By Proposition 4.9(iii), there is a locally free sheaf ${\mathcal{E}}$ on $\overline{{\mathcal{H}}}_{1,2}(7/10)\times {\mathcal{W}}{\mathcal{C}}_{r-1}$ such that $[V({\mathcal{E}})/\mathbb{G}_{m}]$ is the base of the universal family of tacnodal sproutings over $\overline{{\mathcal{H}}}_{1,2}(7/10)\times {\mathcal{W}}{\mathcal{C}}_{r-1}$ . Indeed, we may take ${\mathcal{E}}$ to be

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }({\mathcal{O}}_{{\mathcal{C}}}(-\unicode[STIX]{x1D70E})/{\mathcal{O}}_{{\mathcal{C}}}(-2\unicode[STIX]{x1D70E})\oplus {\mathcal{O}}_{{\mathcal{C}}}(-\unicode[STIX]{x1D70F})/{\mathcal{O}}_{{\mathcal{C}}}(-2\unicode[STIX]{x1D70F})),\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}:{\mathcal{C}}\rightarrow \overline{{\mathcal{H}}}_{1,2}(7/10)\times {\mathcal{W}}{\mathcal{C}}_{r-1}$ is the universal family, $\unicode[STIX]{x1D70E}$ corresponds to one of the universal sections over $\overline{{\mathcal{H}}}_{1,2}(7/10)$ , and $\unicode[STIX]{x1D70F}$ corresponds to the universal section over ${\mathcal{W}}{\mathcal{C}}_{r-1}$ . If ${\mathcal{V}}\subset [V({\mathcal{E}})/\mathbb{G}_{m}]$ is the open locus parameterizing sproutings which do not introduce an elliptic bridge, then ${\mathcal{V}}$ is the complement of the sub-bundle $[V(\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{C}}}(-\unicode[STIX]{x1D70F}))/\mathbb{G}_{m}]\subset [V({\mathcal{E}})/\mathbb{G}_{m}]$ . Since $\overline{{\mathcal{H}}}_{1,2}(7/10)\times {\mathcal{W}}{\mathcal{C}}_{r-1}$ admits a good moduli space, and $V({\mathcal{E}})\backslash V(\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{{\mathcal{C}}}(-\unicode[STIX]{x1D70F}))$ is affine over $\overline{{\mathcal{H}}}_{1,2}(7/10)\times {\mathcal{W}}{\mathcal{C}}_{r-1}$ , we have that ${\mathcal{V}}$ admits a good moduli space. By Lemma 4.2, there is an induced morphism $\unicode[STIX]{x1D6F9}:{\mathcal{V}}\rightarrow \overline{{\mathcal{M}}}_{2r,1}(2/3).$ The image of $\unicode[STIX]{x1D6F9}$ is ${\mathcal{W}}{\mathcal{C}}_{r}$ , and $\unicode[STIX]{x1D6F9}$ is finite by Proposition 4.7. Thus, Proposition 1.4 implies that ${\mathcal{W}}{\mathcal{C}}_{r}$ admits a good moduli space.◻

Proof. The morphism $\unicode[STIX]{x1D6F9}$ is well defined by Lemma 4.1. To see that $\unicode[STIX]{x1D6F9}$ is finite, first observe that $\unicode[STIX]{x1D6F9}$ is clearly representable and quasi-finite. Furthermore, since the limit of a disconnecting node is a disconnecting node in $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Corollary 2.11], $\unicode[STIX]{x1D6F9}$ satisfies the valuative criterion for properness.◻

Proof. These gluing maps are well defined by Lemma 4.1, and finiteness follows as in Lemma 4.13. ◻

Proof. Let $\unicode[STIX]{x1D6FC}_{c}=9/11$ . By Lemma 4.11, we may assume that $(g,n)\neq (1,1)$ . By Lemma 4.13, there exists a finite gluing morphism

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}:\overline{{\mathcal{M}}}_{g-1,n+1}(9/11)\times \overline{{\mathcal{H}}}_{1,1}(9/11)\rightarrow \overline{{\mathcal{M}}}_{g,n}(9/11),\end{eqnarray}$$

whose image is precisely $\overline{{\mathcal{H}}}_{g,n}(9/11)$ . Now $\overline{{\mathcal{H}}}_{g,n}(9/11)$ admits a proper good moduli space by Proposition 1.4.

Let $\unicode[STIX]{x1D6FC}_{c}=7/10$ . For every $r$ such that $\overline{{\mathcal{M}}}_{g-2r,n+2}(7/10)$ (respectively, $\overline{{\mathcal{M}}}_{g-2r-1,n}(7/10)$ ) exists, Lemma 4.14 (respectively, Lemma 4.13) gives a finite gluing morphism

$$\begin{eqnarray}\displaystyle & \displaystyle \overline{{\mathcal{M}}}_{g-2r,n+2}(7/10)\times {\mathcal{E}}{\mathcal{C}}_{r}\rightarrow \overline{{\mathcal{H}}}_{g,n}(7/10) & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\text{respectively, }\overline{{\mathcal{M}}}_{g-2r-1,n}(7/10)\times {\mathcal{E}}{\mathcal{C}}_{r}\rightarrow \overline{{\mathcal{H}}}_{g,n}(7/10)) & \displaystyle \nonumber\end{eqnarray}$$

that identifies $(C,\{p_{i}\}_{i=1}^{n+2})$ (respectively, $(C,\{p_{i}\}_{i=1}^{n})$ ) to $(Z,q_{1},q_{2})$ at $p_{n+1}\sim q_{1}$ , $p_{n+2}\sim q_{2}$ (respectively, $p_{n}\sim q_{1}$ ). In addition, for every triple of integers $(i,m,r)$ such that the stack $\overline{{\mathcal{M}}}_{i,m+1}(7/10)\times \overline{{\mathcal{M}}}_{g-i-2r+1,n-m+1}(7/10)$ is defined, Lemma 4.13 gives a finite gluing morphism

$$\begin{eqnarray}\displaystyle \overline{{\mathcal{M}}}_{i,m+1}(7/10)\times \overline{{\mathcal{M}}}_{g-i-2r+1,n-m+1}(7/10)\times {\mathcal{E}}{\mathcal{C}}_{r}\rightarrow \overline{{\mathcal{H}}}_{g,n}(7/10), & & \displaystyle \nonumber\end{eqnarray}$$

which identifies $(C,\{p_{i}\}_{i=1}^{m+1})$ , $(C^{\prime },\{p_{i}^{\prime }\}_{i=1}^{n-m+1})$ , $(Z,q_{1},q_{2})$ nodally at $p_{m+1}\sim q_{1}$ , $p_{n-m+1}^{\prime }\sim q_{2}$ . Finally, if $(g,n)=(2r,0)$ , Lemma 4.14 gives a finite gluing morphism

$$\begin{eqnarray}\displaystyle {\mathcal{E}}{\mathcal{C}}_{r}\rightarrow \overline{{\mathcal{H}}}_{2r}(7/10), & & \displaystyle \nonumber\end{eqnarray}$$

which nodally self-glues $(Z,q_{1},q_{2})$ at $q_{1}\sim q_{2}$ . The union of these gluing morphisms covers $\overline{{\mathcal{H}}}_{g,n}(7/10)$ . Thus, $\overline{{\mathcal{H}}}_{g,n}(7/10)$ admits a proper good moduli space by Proposition 1.4 and Lemma 4.12.

Let $\unicode[STIX]{x1D6FC}_{c}=2/3$ . By Lemma 4.11, we may assume that $(g,n)\neq (2,1)$ . For each $r=1,\ldots ,\lfloor g/2\rfloor$ , Lemma 4.13 provides a finite gluing morphism

$$\begin{eqnarray}\overline{{\mathcal{M}}}_{g-2r,n+1}(2/3)\times {\mathcal{W}}{\mathcal{C}}_{r}(2/3)\rightarrow \overline{{\mathcal{M}}}_{g,n}(2/3)\end{eqnarray}$$

(if $r=g/2$ and $n=1$ , we consider $\overline{{\mathcal{M}}}_{g-2r,n+1}(2/3)$ as the empty set). The union of these gluing morphisms covers $\overline{{\mathcal{H}}}_{g,n}(2/3)$ . Now $\overline{{\mathcal{H}}}_{g,n}(2/3)$ admits a proper good moduli space by Proposition 1.4 and Lemma 4.12.◻

Proof of Theorem 1.1.

Fix $\unicode[STIX]{x1D6FC}_{c}\in \{9/11,7/10,2/3\}$ . Note that $\overline{{\mathcal{M}}}_{0,n}(\unicode[STIX]{x1D6FC}_{c})=\overline{{\mathcal{M}}}_{0,n}$ , so $\overline{{\mathcal{M}}}_{0,n}(\unicode[STIX]{x1D6FC}_{c})$ admits a proper good moduli space for all $n$ . By induction on $g$ , we may assume that $\overline{{\mathcal{M}}}_{g^{\prime },n^{\prime }}(\unicode[STIX]{x1D6FC}_{c})$ admits a proper good moduli space for all $(g^{\prime },n^{\prime })$ with $g^{\prime }<g$ . Note that $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})=\overline{{\mathcal{M}}}_{g,n}$ for $\unicode[STIX]{x1D6FC}>9/11$ . By descending induction on $\unicode[STIX]{x1D6FC}$ , we may now assume that $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ admits a good moduli space for all $\unicode[STIX]{x1D6FC}\geqslant \unicode[STIX]{x1D6FC}_{c}+\unicode[STIX]{x1D716}$ . By [Reference Alper, Fedorchuk, Smyth and van der WyckAFSvdW17, Theorem 3.17], the inclusions $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}){\hookleftarrow}\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D716})$ arise from local VGIT with respect to $\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D713}$ , and Propositions 4.15 and 4.10 imply that $\overline{{\mathcal{H}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})=\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})\setminus \overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}-\unicode[STIX]{x1D716})$ and $\overline{{\mathcal{S}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})=\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})\setminus \overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}+\unicode[STIX]{x1D716})$ admit proper good moduli spaces. Now Theorem 1.3 implies that $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c})$ and $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC}_{c}-\unicode[STIX]{x1D716})$ admit proper good moduli spaces fitting into the stated diagram.◻