Proof. The statements follow directly from the proof of [Reference Battistella and CarocciBC23, Proposition 3.12], where the boundary components of $\mathbb P\Omega \overline {{\mathcal M}}_{2,1}^{\textrm {inc}}(2)$ are described explicitly.

Proof. Consider the first case when $\delta _1$ and $\delta _2$ are on a tail $R$ , which implies that $R$ is smooth. In this case, we further assume that $F$ is smooth; the sub-case when $F$ is singular follows from the closedness of $\overline {{\mathcal K}}$ and $\overline {{\mathcal W}}$ . By (2.16), we see that $(C,\delta _1,\delta _2) \in \overline {{\mathcal K}}$ if and only if there exists a pointed stable differential of type $(1,1)$ over $(C,\delta _1,\delta _2)$ satisfying the conditions in [Reference Battistella and CarocciBC23, Theorem 1.3]. Notice that on $R$ , to scaling, there uniquely exists a (non-zero) meromorphic differential with simple zeros at $\delta _1$ and $\delta _2$ , as well as a pole at $q := \langle {\delta _1}\rangle$ whose order and residue are, respectively, $-4$ and 0. Analyzing the five conditions of [Reference Battistella and CarocciBC23, Definitions 1.1 & 1.2], we conclude that $(C,\delta _1,\delta _2) \in \overline {{\mathcal K}}$ if and only if there exists a (non-zero) holomorphic differential on $F$ with a zero of order 2 at $q$ (because $F$ is assumed to be smooth), i.e. $h^0(F,K_F - 2q)=1$ , which is equivalent to $(F,q) \in \overline {{\mathcal W}}$ .

Consider the second case when $\delta _1 \in F$ is a on a maximal non-separating bridge $\textrm {B} = \textrm {B}[\mathfrak{p},\mathfrak q]$ . Let $\widehat {T}$ be the (connected) genus 1 subcurve of $F$ that meets $\textrm {B}$ at $\mathfrak{p}$ and $\mathfrak q$ .

• • If $\delta _2 \in \textrm {B}$ , to show $(C,\delta _1,\delta _2) \in \overline {{\mathcal K}}$ , we only need to consider the situation when $\textrm {B}$ and $\widehat {T}$ are smooth (because of the closedness of $\overline {{\mathcal K}}$ ); see Graph (a) of Figure 4 for illustration. In this situation, notice that there exists a (non-zero) meromorphic differential on $\textrm {B}$ with simple zeros at $\delta _1$ and $\delta _2$ as well as poles of order 2 at $\mathfrak{p}$ and $\mathfrak q$ , whose total residue on $\textrm {B}$ is 0. Also notice that every (non-zero) holomorphic differential on $\widehat {T}$ has no zeros. Then, by [Reference Battistella and CarocciBC23, Theorem 1.3], we have $(C,\delta _1,\delta _2) \in \overline {{\mathcal K}}$ .

• • If $\delta _2 \notin \textrm {B}$ , let $T'$ be the (unique) genus 1 inseparable subcurve of $\widehat {T}$ . Notice that every element of $H^0(C,\omega _C(-\delta _1))$ , which is of rank 1, vanishes identically on $C\backslash T'$ . In addition, observe that for every level graph $\overline {\Gamma }$ (in the sense of [Reference Battistella and CarocciBC23, Section 1.5]) whose maxima are the irreducible components of $T'$ , every twisted differential $\eta$ compatible with $\overline {\Gamma }$ (in the sense of [Reference Battistella and CarocciBC23, Definitions 1.1 & 1.2]) that has a simple zero at $\delta _1$ must have another simple zero on $\textrm {B}$ . This implies $\eta (\delta _2) \ne 0$ , so by [Reference Battistella and CarocciBC23, Theorem 1.3], we have $(C,\delta _1,\delta _2) \notin \overline {{\mathcal K}}$ .

The statements when $\delta _1 \in F$ is not on any non-separating bridge can be proved by applying [Reference Battistella and CarocciBC23, Theorem 1.3] analogously (on a case-by-case basis). We omit further details.

Proof. Let $Z \subset \overline {{\mathcal K}}$ be the substack consisting of $(C,\delta _1,\delta _2) \in \overline {{\mathcal K}}$ so that there exists a non-separating bridge of (the core of) $C$ containing $\langle {\delta _1}\rangle$ and $\langle {\delta _2}\rangle$ . Similarly, let $Z' \subset \overline {{\mathcal W}}$ be the substack consisting of $(C,\delta _1) \in \overline {{\mathcal W}}$ so that $\delta _1$ is on a non-separating bridge. Then, by Lemmas 2.13 and 2.14,

\begin{align*} Z\simeq \overline {{\mathcal M}}_{1,2} \times \overline {{\mathcal M}}_{0,4}\quad \textrm {and}\quad Z'\simeq \overline {{\mathcal M}}_{1,2} \times \overline {{\mathcal M}}_{0,3}, \end{align*}

so both $Z$ and $Z'$ are closed.

Graph (a) of Figure 4 shows a general point of $Z$ . When one of the marked points approaches a node or the other marked point, we obtain Graph (b) or Graph (c) of Figure 4. In addition, the genus 1 subcurve $\widehat {T}$ in the first row of Figure 4 may be nodal; in particular, $\widehat {T}$ may even contain a separating node, as shown in the second row of Figure 4.

By Lemma 2.14, the restriction to $\overline {{\mathcal K}}$ of the forgetful morphism $\overline {{\mathcal M}}_{2,2} \longrightarrow \overline {{\mathcal M}}_{2,1}$ that forgets the second marked point, still denoted by

\begin{align*}\widehat {{\rm pr\,} }:\overline {{\mathcal K}}\longrightarrow \overline {{\mathcal M}}_{2,1},\end{align*}

restricts to an isomorphism of the open substacks

\begin{align*} \overline {{\mathcal K}}\backslash Z\longrightarrow \overline {{\mathcal M}}_{2,1}\backslash Z'. \end{align*}

Hence $\overline {{\mathcal K}}\backslash Z$ is smooth and of codimension 1 in $\overline {{\mathcal M}}_{2,2}\backslash Z$ . Therefore, to establish Proposition 2.15, it suffices to fix an arbitrary point

\begin{align*}\widehat {x}=(\widehat {C},\widehat {\delta }_{1},\widehat {\delta }_{2})\in Z\end{align*}

as well as a small affine smooth chart ${\mathcal V} \longrightarrow \overline {{\mathcal M}}_{2,2}$ of $x$ , and show that $\overline {{\mathcal K}} \cap {\mathcal V}$ is a smooth Cartier divisor of $\mathcal V$ .

Below, we prove this by analyzing $\kappa \in \Gamma ({\mathscr O}_{\mathcal V})$ with

(2.18) \begin{align} \overline {{\mathcal K}}\cap {\mathcal V}=\{\kappa=0\}. \end{align}

To this end, we denote by $\textrm {B}[\mathfrak{p},\mathfrak q]$ the maximal non-separating bridge of $\widehat {C}$ containing $\langle {\widehat {\delta }_{1}}\rangle$ and $\langle {\widehat {\delta }_{2}}\rangle$ , and by $\widehat {T}$ the genus 1 connected subcurve of $\widehat {C}$ meeting $\textrm {B}[\mathfrak{p},\mathfrak q]$ at $\mathfrak{p}$ and $\mathfrak q$ ; see Figure 4. Observe that

\begin{align*} Z\cap {\mathcal V}=\{\zeta _{\mathfrak{p}} = \zeta _{\mathfrak q} = 0\}. \end{align*}

This implies that $\kappa=0$ whenever $\zeta _{\mathfrak{p}} = \zeta _{\mathfrak q} = 0$ , so there exist $f,g \in \Gamma ({\mathscr O}_{{\mathcal V}})$ such that

(2.19) \begin{align} \kappa =f\zeta _{\mathfrak{p}}+g\zeta _{\mathfrak q}. \end{align}

We aim to show that both $f$ and $g$ are invertible. By symmetry, we only need to show that either of them is invertible. The proof of this is divided into six cases as per Figure 4.

Case (a). In this case, we assume that $\widehat {x}$ is given by Graph (a) of Figure 4, where $\widehat {T}$ may have (up to two) non-separating nodes. Recall that the Hodge bundle $\Omega {\mathcal M}_{2,2}$ extends over $\overline {{\mathcal M}}_{2,2}$ as a rank 2 vector bundle $\Omega \overline {{\mathcal M}}_{2,2}$ , whose fiber over every $(C,\delta _1,\delta _2) \in \overline {{\mathcal M}}_{2,2}$ parameterizes meromorphic differentials $\eta$ of $C$ that have, at worst, simple poles at the nodes of $C$ with opposite residues on the two branches of each node (i.e. the fiber is $H^0(C,\omega _C)$ ).

Since the chart $\mathcal V$ containing $\widehat {x}$ is assumed to be small, the marked points are not on a tail or a separating bridge for any point of $\mathcal V$ . Therefore, for every point $(C,\delta _1,\delta _2) \in {\mathcal V}$ , $\{\eta \in H^0(C,\omega _C):\eta |_{\delta _1} = 0\}$ is always of rank 1. In other words,

\begin{align*} \mathfrak L_1/{\mathcal V}:= \big \{(C,\delta _1,\delta _2,\eta )\in \Omega \overline {{\mathcal M}}_{2,2}/{\mathcal V}:\, \eta |_{\delta _1} = 0\big \} \end{align*}

is a line bundle. We choose a nowhere vanishing section $\mathfrak h_1$ of $\mathfrak L_1/{\mathcal V}$ . In addition, let $L_2/\overline {{\mathcal M}}_{2,2}$ be the universal tangent line bundle at the second marked point. We choose a nowhere vanishing section $h_2$ of $L_2/{\mathcal V}$ . They together give rise to a regular function $\gamma \in \Gamma ({\mathscr O}_{\mathcal V})$ as follows:

(2.20) \begin{align} \gamma (C,\delta _1,\delta _2)=\mathfrak h_1(C,\delta _1,\delta _2)|_{\delta _2}\otimes h_2(C,\delta _1,\delta _2)\quad \forall \,(C,\delta _1,\delta _2) \in {\mathcal V}. \end{align}

Here, $T_{\delta _2}^*C \otimes T_{\delta _2}C$ is canonically identified with $\mathbb C$ .

With $\kappa$ as in (2.19), we claim that

(2.21) \begin{align} \gamma /\kappa \in \Gamma ({\mathscr O}_{\mathcal V}). \end{align}

Indeed, for every

\begin{align*} x=(C,\delta _1,\delta _2)\in \overline {{\mathcal K}}\cap \big ({\mathcal V}\big \backslash \{\zeta _{\mathfrak{p}} \cdot \zeta _{\mathfrak{q}} = 0\}\big ), \end{align*}

by Definition 2.11, we have $\mathfrak h_1(x)|_{\delta _2} = 0$ , so $\gamma /\kappa$ is regular on ${\mathcal V}\big \backslash \{\zeta _{\mathfrak{p}} \cdot \zeta _{\mathfrak{q}} = 0\}$ . Moreover, shrinking $\mathcal V$ if necessary, we have

\begin{align*} \overline {{\mathcal K}}\cap {\mathcal V}\cap \big (\{\zeta _{\mathfrak{p}} \ne 0,\zeta _{\mathfrak q} = 0\} \cup \{\zeta _{\mathfrak q} \ne 0,\zeta _{\mathfrak{p}} = 0\}\big )=\emptyset ,\end{align*}

for otherwise, without loss of generality we suppose that there exists a sequence of elements of $\overline {{\mathcal K}} \cap \{\zeta _{\mathfrak{p}} \ne 0,\,\zeta _{\mathfrak q} = 0\}$ approaching to $\widehat {x}$ . Then, by [Reference Battistella and CarocciBC23, Theorem 1.3], $H^0(\widehat {T},\omega _{\widehat {T}})$ must have a nonzero element vanishing at $\mathfrak{p}$ . As this is impossible, we see that $\gamma /\kappa$ is regular on the complement of the codimension 2 substack $Z \cap {\mathcal V}$ , and hence on the entire $\mathcal V$ by Hartogs’ Lemma.

Next, we construct a morphism of a small disk $D \longrightarrow {\mathcal V}$ whose image contains $\widehat {x}$ , and analyze the vanishing order of the pullback of $\gamma$ at the center of $D$ . More precisely, we write

\begin{align*} \widehat {\eta }^+:=\mathfrak h_1(\widehat {x})\quad \big (\in H^0\big (\widehat {C},\omega _{\widehat {C}}\big )\backslash \{0\}\big ). \end{align*}

Then $\widehat {\eta }^+$ vanishes identically 0 on $\textrm {B}[\mathfrak{p},\mathfrak q]$ , which in Case (a) is a smooth rational subcurve of $C$ , denoted by $\widehat {R}$ . On the genus 1 inseparable component $\widehat {T}$ , the restriction of $\widehat {\eta }^+$ is holomorphic, hence does not have any zero. Next, we choose another

\begin{align*} \widehat {\xi }\in H^0\big (\widehat {C},\omega _{\widehat {C}}\big ) \quad \textrm {s.t.}\quad H^0\big (\widehat {C},\omega _{\widehat {C}}\big )=\textrm {span}\big \{\widehat {\eta }^+,\,\widehat {\xi }\big \}.\end{align*}

Then, $\widehat {\xi }$ does not have any zero on $\widehat {R}$ . Moreover, $\widehat {\xi }$ has simple poles at $\mathfrak{p}$ and $\mathfrak q$ , such that on both $\widehat {T}$ and $\widehat {R}$ , the residues of $\widehat {\xi }$ sum up to 0. Finally, there exists a unique

\begin{align*} \widehat {\eta }^{\,-}\in H^0\big (\widehat {R}, K_{\widehat {R}} + 2\mathfrak{p}\big )\quad \textrm {s.t.}\quad \big (\widehat {\eta }^{\,-} + \widehat {\xi }\big )|_{\widehat {\delta }_{1}} = 0.\end{align*}

Notice that $\big (\widehat {\eta }^{\,-} + \widehat {\xi }\big )|_{\widehat {R}}$ does not have any other zero on $\widehat {R}$ , because its only poles are $\mathfrak{p}$ (of order 2) and $\mathfrak q$ (of order 1). Consequently, we have

(2.22) \begin{align} (\widehat {\eta }^{\,-} +\widehat {\xi }\,)|_{\widehat {\delta }_{2}}\ne 0. \end{align}

Next, we smooth out $\mathfrak{p}$ while keeping the node $\mathfrak q$ , and apply [Reference Battistella and CarocciBC23, Theorem 4.5 (Higher-order Plumbing)]. Concretely, we take $D$ to be a small disk. By taking

\begin{align*} \eta _t^+ =\widehat {\eta }^+ +t\widehat {\xi },\quad \eta _t^- = t\big (\widehat {\eta }^{\,-} +\widehat {\xi }\big ),\quad t \in D, \end{align*}

in [Reference Battistella and CarocciBC23, Theorem 4.5], we obtain a family of stable curves with two marked points

\begin{align*}x_t=\big (C_t,\delta _{1;t},\delta _{2;t}\big ),\quad t \in D\end{align*}

satisfying:

• • (the central fiber): $x_0=\widehat {x}$ $\big ( = (\widehat {C},\widehat {\delta }_1,\widehat {\delta }_2)\big )$ ;

• • (the nodes): $\zeta _{\mathfrak{p}}(x_t) \ne 0$ for all $t \in D\backslash \{0\}$ , and $\zeta _{\mathfrak{q}}(x_t)=0$ for all $t \in D$ ; and

• • (the vanishing order of $\gamma (x_t)$ ): there exists $u \in \Gamma ({\mathscr O}^*_D)$ such that for every $t \in D,$

  \begin{align*} \gamma (x_t)= t\cdot u(t)\cdot \big ((\widehat {\eta }^{\,-} +\widehat {\xi }\,)\big |_{\widehat {\delta }_{2}}\otimes h_2(\widehat {x})\big ). \end{align*}

Thus, on the one hand, by (2.22), $\gamma (x_t)/t$ is invertible on $D$ . On the other hand, shrinking $D$ if necessary, we see there exist unique integers $\widehat {n} \ge 1$ and $\widehat {m} \ge 0$ such that $\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}$ and $f(x_t)/t^{\widehat {m}}$ are both invertible on $D$ , where $f$ is as in (2.19). Taking $\zeta _{\mathfrak q}(x_t)=0$ for all $t \in D$ into consideration, we conclude from (2.19) and (2.21) that

\begin{align*} \gamma (x_t)/t^{\widehat {n}+\widehat {m}}= \big (\gamma (x_t)/\kappa (x_t)\big )\cdot \big (f(x_t)/t^{\widehat {m}}\big )\cdot \big (\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}\big ) \ \in \ \Gamma ({\mathscr O}_D). \end{align*}

Therefore, $\widehat {n} = 1$ and $\widehat {m} = 0$ . In particular, we have $f(\widehat {x}) \ne 0$ , i.e. $f$ is invertible on the small chart $\mathcal V$ . This completes the proof in Case (a).

Case (b). In this case, we assume that $\widehat {x}$ is given by Graph (b) of Figure 4, where $\widehat {T}$ is separable and may have (at most) one non-separating node. The stability of $\widehat {x}$ implies that the separating node of $\widehat {T}$ is unique, and we denote this by $\mathfrak s$ . Let $\widehat {T}'$ and $\widehat {T}''$ be the two genus 1 inseparable components of $\widehat {C}$ satisfying

\begin{align*}\widehat {T}' \cap \widehat {T}''=\{\mathfrak s\}\quad \textrm {and}\quad \widehat {\delta }_1,\widehat {\delta }_2\in \widehat {T}''.\end{align*}

Then $\widehat {T}''$ consists of two smooth rational subcurves $\widehat {R}$ and $\widehat {R}_{\mathfrak{s}}$ such that

\begin{align*} \widehat {R}=\textrm {B}[\mathfrak{p},\mathfrak q]\quad \textrm {and}\quad \mathfrak s\in \widehat {R}_{\mathfrak{s}}. \end{align*}

The construction of the line bundle $\mathfrak L_1/{\mathcal V}$ of Case (a) is still valid in Case (b), so we define $\gamma \in \Gamma ({\mathscr O}_{\mathcal V})$ in the same way as in (2.20). Notice that for every $x = (C,\delta _1,\delta _2) \in \{\zeta _{\mathfrak{s}} = 0\}$ , $\mathfrak h_1(x)$ vanishes identically on the genus 1 inseparable component of $C$ that contains $\delta _1$ and $\delta _2$ , and hence $\gamma$ vanishes along $\{\zeta _{\mathfrak{s}} = 0\}$ , i.e.

\begin{align*} \gamma /\zeta _{\mathfrak{s}} \in \Gamma ({\mathscr O}_{\mathcal V}). \end{align*}

Moreover, with $\kappa$ as in (2.19), we claim that

(2.23) \begin{align} \gamma \big /\big (\zeta _{\mathfrak{s}} \cdot \kappa \big )\in \Gamma ({\mathscr O}_{\mathcal V}). \end{align}

Indeed, Case (a) implies that $\gamma /(\zeta _{\mathfrak{s}}\kappa )$ is regular on ${\mathcal V}\backslash \{\zeta _{\mathfrak{s}} = 0\}$ . Moreover, for a general point $y \in \{\zeta _{\mathfrak{s}} = 0\}$ , the two marked points of $y$ are not conjugate, by Lemma 2.14, hence $\kappa (y) \ne 0$ , which means that $\gamma /(\zeta _{\mathfrak{s}}\kappa )$ is also regular near every general point of $\{\zeta _{\mathfrak{s}} = 0\}$ . By Hartogs’ Lemma, we obtain (2.23).

Next, we choose the meromorphic differentials $\widehat {\eta }^+$ , $\widehat {\xi }$ , and $\widehat {\eta }^{\,-}$ in the same way as in Case (a). In particular, (2.22) still holds. In Case (b), $\widehat {\eta }^+$ vanishes identically on $\widehat {R}_{\mathfrak{s}}$ . We choose

\begin{align*} \widehat {\eta }_{\mathfrak{s}}\in H^0\big (\widehat {R}_{\mathfrak{s}},K_{\widehat {R}_{\mathfrak{s}}} + 2\mathfrak s\big )\big \backslash \{0\}, \end{align*}

which is unique up to scaling.

Next, we smooth out $\mathfrak s$ and $\mathfrak{p}$ while keeping the node $\mathfrak q$ , and apply the Higher-order Plumbing theorem of [Reference Battistella and CarocciBC23] as in Case (a). Let $D$ be a small disk. In [Reference Battistella and CarocciBC23, Theorem 4.5], we take

\begin{align*} \eta _t^+ =\widehat {\eta }^+,\quad \eta _t^- = t\big (\widehat {\eta }_{\mathfrak{s}} +t\widehat {\xi }\big ),\quad t \in D \end{align*}

for the node $\mathfrak s$ , and

\begin{align*} \ \eta _t^+ = t\big (\widehat {\eta }_{\mathfrak{s}} +t\widehat {\xi }\big ),\quad \eta _t^- = t^2\big (\widehat {\eta }^{\,-} +\widehat {\xi }\big ),\quad t \in D \end{align*}

for the node $\mathfrak{p}$ . Then, [Reference Battistella and CarocciBC23, Theorem 4.5] gives rise to a family of stable curves with two marked points

\begin{align*}x_t=(C_t,\delta _{1;t},\delta _{2;t}),\quad t \in D\end{align*}

over a small disk $D$ , satisfying:

• • (the central fiber): $x_0=\widehat {x}$ $\big ( = (\widehat {C},\widehat {\delta }_1,\widehat {\delta }_2)\big )$ ;

• • (the nodes): $\zeta _{\mathfrak{s}}(x_t) \ne 0$ and $\zeta _{\mathfrak{p}}(x_t) \ne 0$ for all $t \in D\backslash \{0\}$ , and $\zeta _{\mathfrak{q}}(x_t)=0$ for all $t \in D$ ; and

• • (the vanishing order of $\gamma (x_t)$ ): there exists $u \in \Gamma ({\mathscr O}^*_D)$ such that for every $t \in D,$

  \begin{align*} \gamma (x_t)= t^2\cdot u(t)\cdot \big ((\widehat {\eta }^{\,-} +\widehat {\xi })|_{\widehat {\delta }_{2}}\otimes h_2(\widehat {x})\big ). \end{align*}

Thus, on the one hand, by (2.22), $\gamma (x_t)/t^2$ is invertible on $D$ . On the other hand, shrinking $D$ if necessary, we see there exist unique integers $\widehat {k} \ge 1$ , $\widehat {n} \ge 1$ , and $\widehat {m} \ge 0$ such that $\zeta _{\mathfrak s}(x_t)/t^{\widehat {k}}$ , $\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}$ , and $f(x_t)/t^{\widehat {m}}$ are all invertible on $D$ , where $f$ is as in (2.19). Taking into consideration that $\zeta _{\mathfrak q}(x_t)=0$ for all $t \in D$ , we conclude from (2.19) and (2.23) that

\begin{align*} \gamma (x_t)/t^{\widehat {k}+\widehat {n}+\widehat {m}}= (\gamma (x_t)\big /\big (\zeta _{\mathfrak s}(x_t)\kappa (x_t)\big )) \cdot \big (\zeta _{\mathfrak s}(x_t)/t^{\widehat {k}}\big )\cdot \big (f(x_t)/t^{\widehat {m}}\big )\cdot \big (\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}\big ) \,\in \, \Gamma ({\mathscr O}_D). \end{align*}

Therefore, $\widehat {k} = \widehat {n} = 1$ and $\widehat {m} = 0$ . As in Case (a), this implies that $f$ is invertible on the small chart $\mathcal V$ and completes the proof of Case (b).

Cases (c) and (d). In Case (c) (respectively, Case (d)), we assume that $\widehat {x}$ is given by Graph (c) (respectively, Graph (d)) of Figure 4, where $\widehat {T}$ may contain non-separating nodes (respectively, $\widehat {T}$ is separable and may have a non-separating node). The proof of Case (c) (respectively, Case (d)) is indeed parallel to that of Case (a) (respectively, Case (b)), except that $\textrm {B}[\mathfrak{p},\mathfrak q]$ consists of two rational subcurves $\widehat {R}_1$ and $\widehat {R}_2$ respectively containing $\widehat {\delta }_1$ and $\widehat {\delta }_2$ . Without loss of generality we assume $\mathfrak{p} \in \widehat {R}_1$ and $\mathfrak q \in \widehat {R}_2$ .

In either case, we copy the construction of the line bundle $\mathfrak L_1$ and the 1-forms $\widehat {\eta }^+$ and $\widehat {\xi }$ verbatim, and define $\widehat {\eta }^{\,-}$ likewise on $\widehat {R}_1$ . In addition, in Case (d), we choose $\widehat {\eta }_{\mathfrak{s}}$ on $\widehat {R}_{\mathfrak{s}}$ as in Case (b). Then, in Case (c) (respectively, Case (d)), we still have (2.22), and hence [Reference Battistella and CarocciBC23, Theorem 4.5] still gives rise to a family $\{x_t\} \subset {\mathcal V}$ satisfying the same properties as in Case (a) (respectively, Case (b)). Hence the argument in Case (a) (respectively, Case (b)) still applies to Case (c) (respectively, Case (d)).

Case (e). In this case, we assume that $\widehat {x}$ is given by Graph (e) of Figure 4, where $\widehat {T}$ may have non-separating nodes. In particular, $\widehat {\delta }_1$ and $\widehat {\delta }_2$ are on a smooth tail $\widehat {R}_{\mathfrak r}$ that attaches to $\widehat {R} := \textrm {B}[\mathfrak{p},\mathfrak q]$ at the node $\mathfrak r$ .

On the small chart $\mathcal V$ containing $\widehat {x}$ ,

\begin{align*} \mathfrak L_1'/{\mathcal V}:= \big \{(C,\delta _1,\delta _2,\eta )\in \Omega \overline {{\mathcal M}}_{2,2}|_{{\mathcal V}}:\, \eta |_{\langle {\delta _1}\rangle } = 0\big \} \end{align*}

is a line bundle, on which we choose a nowhere vanishing section $\mathfrak h_1'$ . Let $L_2/{\mathcal V}$ still be the universal tangent line bundle at the second marked point, on which we choose a nowhere vanishing section $h_2$ .

Consider the regular function $\gamma ^{\,\prime } \in \Gamma ({\mathscr O}_{\mathcal V})$ given by

\begin{align*} \gamma ^{\,\prime }(C,\delta _1,\delta _2)=\mathfrak h_1'(C,\delta _1,\delta _2)|_{\delta _2}\otimes h_2(C,\delta _1,\delta _2)\quad \forall \,(C,\delta _1,\delta _2) \in {\mathcal V}. \end{align*}

First, we claim that

(2.24) \begin{align} \gamma :=\gamma ^{\,\prime }\big /\zeta _{\mathfrak r}^2\in \Gamma ({\mathscr O}_{\mathcal V}), \end{align}

where $\zeta _{\mathfrak r}$ is a node-smoothing parameter corresponding to the node $\mathfrak r$ . To see this, we denote by $\Delta _{12} \subset \overline {{\mathcal M}}_{2,2}$ the (divisorial) boundary component of $\overline {{\mathcal M}}_{2,2}$ when $\delta _1$ and $\delta _2$ are on the same tail. Locally, we have $\Delta _{12} \cap {\mathcal V} = \{\zeta _{\mathfrak r} = 0\}.$ Note that

\begin{align*}L_2\simeq \widehat {{\rm pr\,} }_2^*L\otimes {\mathscr O}(-\Delta _{12}),\end{align*}

where $\widehat {{\rm pr\,} }_2 :\overline {{\mathcal M}}_{2,2} \longrightarrow \overline {{\mathcal M}}_{2,1}$ denotes the forgetful morphism that forgets the first marked point, and $L/\overline {{\mathcal M}}_{2,1}$ is the universal tangent line bundle at the unique marked point. Taking into consideration the fact that $\mathfrak h_1'(C,\delta _1,\delta _2)|_{\langle {\delta _2}\rangle }$ vanishes on $\Delta _{12} \cap {\mathcal V}$ (which follows immediately from the definition of $\mathfrak L_1'$ ), we see that the vanishing order of $\gamma ^{\,\prime }$ along $\Delta _{12} \cap {\mathcal V}$ is at least 2, and thus we obtain (2.24).

Next, we claim that this defined $\gamma$ vanishes on $\overline {{\mathcal K}}$ ; in other words, with $\kappa$ as in (2.19), we have

(2.25) \begin{align} \gamma ^{\,\prime }\big /\big (\zeta _{\mathfrak r}^2 \cdot \kappa \big )\in \Gamma ({\mathscr O}_{\mathcal V}). \end{align}

To see this, first observe that Case (a) implies that $\gamma /\kappa$ is regular on ${\mathcal V}\backslash \Delta _{12}$ . In addition, for a general point $y \in \Delta _{12} \cap {\mathcal V}$ , the node $\mathfrak r$ is not a Weierstrass point on the core of $F$ . Hence, by Lemma 2.14, $\kappa (y) \ne 0$ , which means that $\gamma /\kappa$ is regular near every general point of $\Delta _{12} \cap {\mathcal V}$ . Once again by Hartogs’ Lemma, we obtain (2.25).

Parallel to Case (a), we set

\begin{align*} \widehat {\eta }^+:=\mathfrak h_1'(\widehat {x})\quad \big (\in H^0\big (\widehat {C},\omega _{\widehat {C}}\big )\backslash \{0\}\big ) \end{align*}

and take $\widehat {\xi }$ and $\widehat {\eta }^{\,-}$ in the same way as in Case (a). In addition, we take $\widehat {\eta }_{\mathfrak r}$ to be a (unique up to scaling) nonzero meromorphic differential on $\widehat {R}_{\mathfrak r}$ whose only pole and zero are an order 3 pole at $\mathfrak r$ and a simple zero at $\widehat {\delta }_1$ , respectively. In particular,

(2.26) \begin{align} \widehat {\eta }_{\mathfrak r}|_{\widehat {\delta }_2}\ne 0. \end{align}

Next, we smooth out $\mathfrak{p}$ and $\mathfrak r$ while keeping the node $\mathfrak q$ , and apply the Higher-order Plumbing theorem of [Reference Battistella and CarocciBC23] again. Let $D$ be a small disk. In [Reference Battistella and CarocciBC23, Theorem 4.5], we take

\begin{align*} \eta _t^+ =\widehat {\eta }^+ + t\widehat {\xi },\quad \eta _t^- = t\big (\widehat {\eta }^{\,-} +\widehat {\xi }\big ),\quad t \in D \end{align*}

for the node $\mathfrak{p}$ , and

\begin{align*} \eta _t^+ = t\big (\widehat {\eta }^{\,-} +\widehat {\xi }\big ),\quad \eta _t^- = t^3\widehat {\eta }_{\mathfrak r},\quad t \in D \end{align*}

for the node $\mathfrak r$ . Then, [Reference Battistella and CarocciBC23, Theorem 4.5] gives rise to a family of stable curves with two marked points

\begin{align*}x_t=(C_t,\delta _{1;t},\delta _{2;t}),\quad t \in D\end{align*}

over a small disk $D$ , satisfying:

• • (the central fiber): $x_0=\widehat {x}$ $\big ( = (\widehat {C},\widehat {\delta }_1,\widehat {\delta }_2)\big )$ ;

• • (the nodes): $\zeta _{\mathfrak{p}}(x_t) \ne 0$ and $\zeta _{\mathfrak r}(x_t) \ne 0$ for all $t \in D\backslash \{0\}$ , and $\zeta _{\mathfrak{q}}(x_t)=0$ for all $t \in D$ ; and

• • (the vanishing order of $\gamma ^{\,\prime }(x_t)$ ): there exists $u \in \Gamma ({\mathscr O}^*_D)$ such that for every $t \in D,$

  \begin{align*} \gamma ^{\,\prime }(x_t)= t^3\cdot u(t)\cdot \big (\widehat {\eta }_{\mathfrak r}|_{\widehat {\delta }_{2}}\otimes h_2(\widehat {x})\big ). \end{align*}

Thus, on the one hand, by (2.26), $\gamma ^{\,\prime }(x_t)/t^3$ is invertible on $D$ . On the other hand, shrinking $D$ if necessary, we see there exist unique integers $\widehat {n} \ge 1$ , $\widehat {k} \ge 1$ , and $\widehat {m} \ge 0$ such that $\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}$ , $\zeta _{\mathfrak r}(x_t)/t^{\widehat {k}}$ , and $f(x_t)/t^{\widehat {m}}$ are all invertible on $D$ , where $f$ is as in (2.19). Taking into consideration that $\zeta _{\mathfrak q}(x_t)=0$ for all $t \in D$ , we conclude from (2.19) and (2.25) that

\begin{align*} \gamma ^{\,\prime }(x_t)/t^{2\widehat {k}+\widehat {n}+\widehat {m}}= (\gamma ^{\,\prime }(x_t)/(\zeta^2_{\mathfrak r}(x_t)\kappa (x_t))) \cdot \big (\zeta _{\mathfrak r}(x_t)/t^{\widehat {k}}\big )^2\cdot \big (f(x_t)/t^{\widehat {m}}\big )\cdot \big (\zeta _{\mathfrak{p}}(x_t)/t^{\widehat {n}}\big ) \,\in \, \Gamma ({\mathscr O}_D). \end{align*}

Therefore, $\widehat {k} = \widehat {n} = 1$ and $\widehat {m} = 0$ . As in Case (a), this implies that $f$ is invertible on the small chart $\mathcal V$ and completes the proof of Case (e).

Case (f). In this case, we assume that $\widehat {x}$ is given by Graph (f) of Figure 4, where $\widehat {T}$ is separable and may have a non-separating node.

The proof of this case simply combines the arguments used in Cases (b) and (e). Specifically, we construct the line bundle $\mathfrak L_1'/{\mathcal V}$ and the regular function $\gamma ^{\,\prime }$ as in Case (e), and observe that

\begin{align*} \gamma ^{\,\prime }\big /\big (\zeta _{\mathfrak r}^2 \cdot \zeta _{\mathfrak{s}} \cdot \kappa \big )\in \Gamma ({\mathscr O}_{\mathcal V}), \end{align*}

where $\mathfrak s$ denotes the separating node of the core and $\mathfrak r$ denotes the pivotal node where the tail attaches to the core. Then, we choose the meromorphic differentials $\widehat {\eta }^+$ , $\widehat {\xi }$ , $\widehat {\eta }^{\,-}$ and $\widehat {\eta }_{\mathfrak r}$ as in Case (e), as well as $\widehat {\eta }_{\mathfrak{s}}$ as in Case (b). Once again, we smooth out $\mathfrak s$ , $\mathfrak{p}$ and $\mathfrak r$ while keeping the node $\mathfrak q$ , and apply [Reference Battistella and CarocciBC23, Theorem 4.5] to obtain a family $\{x_t\} \subset {\mathcal V}$ with $x_0 = \widehat {x}$ such that $\gamma ^{\,\prime }(x_t)/t^4$ is invertible, which implies that $f(x_t)$ is invertible. We omit further details.

Proof. Observe that the first statement of Lemma 2.14 implies an inclusion

\begin{align*} \overline {{\mathcal W}}\simeq \overline {{\mathcal W}}\times \overline {{\mathcal M}}_{0,3}\hookrightarrow \overline {{\mathcal K}} \end{align*}

by identifying the unique marked point of $\overline {{\mathcal W}}$ with the first marked point of $\overline {{\mathcal M}}_{0,3}$ . Given $x = (C,\delta _1) \in \overline {{\mathcal W}}$ , we denote by $\widehat {x} = (\widehat {C},\widehat {\delta }_1,\widehat {\delta }_2)$ its image in $\overline {{\mathcal K}}$ , and by $\mathfrak r$ the node where the unique tail attaches to the core $\widehat {F}$ . On a small affine smooth chart ${\mathcal V} \longrightarrow \overline {{\mathcal M}}_{2,2}$ containing $\widehat {x}$ , the image of $\overline {{\mathcal W}}$ under the above inclusion can be written as

(2.27) \begin{align} \big \{\zeta _{\mathfrak r}=\kappa=0\big \}, \end{align}

where $\kappa$ is the local parameter satisfying $\overline {{\mathcal K}} \cap {\mathcal V} = \{\kappa=0\}$ .

Let $Z \subset \overline {{\mathcal K}}$ be the closed substack of $\overline {{\mathcal K}}$ as in the beginning of the proof of Proposition 2.15.

If $\widehat {x} \in \overline {{\mathcal K}}\backslash Z$ , then $\mathfrak r$ is a Weierstrass point of $\widehat {F}$ away from any separating or non-separating bridge, and hence is a branch point of the hyperelliptic admissible cover. Therefore, at $\widehat {x}$ , the normal direction to $\{\kappa=0\}$ is given by changing the position of $\mathfrak r$ on $\widehat {F}$ , which is transverse to the smoothing of the nodes of $\widehat {C}$ . Hence $\{\kappa \} \sqcup \{\zeta _q:q \in N(\widehat {C})\}$ forms a subset of a system of local parameters on $\mathcal V$ . This implies that (2.27) is smooth on $\mathcal V$ .

If $\widehat {x} \in Z$ , let $\mathfrak{p}$ and $\mathfrak q$ be the core nodes of $\widehat {C}$ determining the (maximal) non-separating bridge containing $\mathfrak r$ . By (2.19), $\kappa = f\zeta _{\mathfrak{p}} + g\zeta _{\mathfrak q}$ for some units $f$ and $g$ on $\mathcal V$ . Since the node-smoothing parameters $\zeta _{\mathfrak{p}}$ , $\zeta _{\mathfrak{q}}$ and $\zeta _{\mathfrak r}$ form a subset of a system of local parameters on $\mathcal V$ , we conclude that (2.27) is smooth on $\mathcal V$ .

Proof of Proposition 2.17. Part (i) holds because $\overline {{\mathcal K}}$ is a Cartier divisor of $\overline {{\mathcal M}}_{2,2}$ by Proposition 2.15.

To verify part (ii), we denote by $\overline {x} = (\overline {C},\overline {\delta }_i,\overline {\delta }_j) \in \overline {{\mathcal K}}$ the image of $x$ by forgetting all but the $i$ th and $j$ th marked points. Let $\overline {{\mathcal V}} \subset \overline {{\mathcal M}}_{2,1}$ be a small affine smooth chart centered at $\overline {x}$ . By the smoothness of $\overline {{\mathcal K}}$ (proved in Proposition 2.15), there exists a local parameter $\overline {\kappa }$ on $\overline {{\mathcal V}}$ such that

\begin{align*} \overline {{\mathcal K}}\cap \overline {{\mathcal V}}=\{\overline {\kappa } = 0\}. \end{align*}

If $\overline {\delta }_i$ (hence $\overline {\delta }_j$ ) is on the core and not on any non-separating bridge, then by Lemma 2.14, for every $(C',\delta _1',\delta _2') \in \{\overline {\kappa } = 0\}$ , the location of $\delta _2'$ on $C'$ is determined by $\delta _1'$ . Therefore, at $\overline {x}$ , the normal direction to $\{\overline {\kappa } = 0\}$ is given by changing the position of $\overline {\delta }_j$ on $\overline {C}$ , which is transverse to the smoothing of the nodes of $\overline {C}$ . Consequently, $\{\overline {\kappa }\} \sqcup \{\zeta _q:q \in N(\overline {C})\}$ forms a subset of a system of local parameters on $\overline {{\mathcal V}}$ . Applying Lemma 2.18 inductively, we see that $\overline {\kappa }$ pulls back to a local parameter $\kappa$ on $\mathcal V$ , satisfying

\begin{align*} \overline {{\mathcal K}}_{\ell ;i,j}\cap {\mathcal V}=\{\kappa=0\}. \end{align*}

In addition, Lemma 2.18 implies that $\{\kappa \} \sqcup \{\zeta _q:q \in N(C)\}$ is a subset of a system of local parameters on $\mathcal V$ .

If $\overline {\delta }_i$ and $\overline {\delta }_j$ are on the same tail $R$ that meets the core $\overline {F}$ of $\overline {C}$ at a pivotal node $\mathfrak r$ , and $\mathfrak r$ is not on any non-separating bridge, then, as shown in the proof of Corollary 2.16, $\{\overline {\kappa }\} \sqcup \{\zeta _q:q \in N(\overline {C})\}$ still forms a subset of a system of local parameters on $\overline {{\mathcal V}}$ , and hence the argument from the previous paragraph applies to this case verbatim.

To verify part (iii), we denote by $\overline {x} = (\overline {C},\overline {\delta }_i,\overline {\delta }_j) \in \overline {{\mathcal M}}_{2,2}$ the image of $x$ by forgetting all but the $i$ th and $j$ th marked points. Let $\overline {{\mathcal V}} \subset \overline {{\mathcal M}}_{2,1}$ be a small affine smooth chart centered at $\overline {x}$ . Assume that $\langle {\overline {\delta }_i}\rangle$ is on a maximal non-separating bridge $\textrm {B}[\overline {\mathfrak{p}},\overline {\mathfrak q}]$ of $\overline {C}$ . Then by Lemmas 2.14 and 2.13, we see that $\overline {x} \in \overline {{\mathcal K}}$ if and only if $\langle {\overline {\delta }_j}\rangle \in \textrm {B}[\overline {\mathfrak{p}},\overline {\mathfrak q}]$ . Moreover, let $\zeta _{\overline {\mathfrak{p}}}$ and $\zeta _{\overline {\mathfrak q}}$ be node-smoothing parameters on $\overline {{\mathcal V}}$ corresponding to $\overline {\mathfrak{p}}$ and $\overline {\mathfrak q}$ , respectively. Then, by the proof of Proposition 2.15 (cf. (2.19)), there exist invertible functions $f$ and $g$ on $\overline {{\mathcal V}}$ such that

\begin{align*} \overline {\kappa }=f\zeta _{\overline {\mathfrak{p}}}+g\zeta _{\overline {\mathfrak q}}. \end{align*}

Applying Lemma 2.18 inductively, we see that $\zeta _{\overline {\mathfrak{p}}}$ and $\zeta _{\overline {\mathfrak q}}$ pull back to

\begin{align*}\prod _{q\in N_{[z_{\langle {\delta _i}\rangle },\mathfrak{p}]}\cap N_{[z_{\langle {\delta _j}\rangle },\mathfrak{p}]}} \zeta _q \quad \textrm {and}\quad \prod _{q\in N_{[z_{\langle {\delta _i}\rangle },\mathfrak q]}\cap N_{[z_{\langle {\delta _j}\rangle },\mathfrak q]}} \zeta _q, \end{align*}

respectively, each up to a unit. This completes part (iii).