text:
\begin{document}

\begin{abstract}
We prove that the moduli of stable supercurves with punctures is a smooth proper DM stack and study an analog of the Mumford's isomorphism for its canonical line bundle.
\end{abstract}

\maketitle

\section{Introduction}

The moduli space $\SS_g$ of supercurves of genus $g$ (aka super Riemann surfaces, aka SUSY curves) has been around in mathematics and physics
since the 80's. It plays an important role in superstring theory and has been studied using the language of algebraic geometry in \cite{LR}, \cite{CV}, \cite{BHR}, \cite{DW} and
other works.
One long standing gap in the mathematical side of the story has been the study of the analog of Deligne-Mumford compactification by stable supercurves
(which is a proper Deligne-Mumford superstack).
One of the goals of the present paper is to contribute to filling this gap. This compactification seems to be necessary for continuing the study of superstring supermeasure (see
\cite{Witten}, \cite{FKP}).

The definition of a stable supercurve and a sketch of a proof that this gives a smooth and proper DM-stack is contained Deligne's letter
to Manin \cite{Deligne} (supernodes were also introduced independently in physics literature, see \cite{Cohn}). The main part of Deligne's letter is
devoted to the infinitesimal part of the theory. In particular, he describes miniversal deformations of two types fo supernode singularities (see
Sec.\ \ref{nodes-def-sec}). The idea of the rest of the proof is to use the superanalog of the Artin's criterion for proving algebraicity of a stack.

In the present paper we revisit Deligne's letter and generalize its results to the case of stable supercurves with punctures.
One of the special features of the super-case is that instead of ``marked points" we consider supercurves with {\it Neveu-Schwarz (NS) and Ramond punctures}.
Of these, NS punctures are like marked points: for a family of (stable) supercurves $X\to S$ they are given by sections $S\to X$. However, Ramond punctures have a different
nature: they are given by relative Cartier divisors $R\sub X$ such that the projection $R\to S$ is smooth of dimension $0|1$, and also the definition of being a supercurve
is modified near $R$ (see Sec.\ \ref{def-stable-supercurve-sec} for details). Note that the nodes of stable supercurves can also be of two types, NS and Ramond.

\bigskip

\noindent
{\bf Theorem A}. {\it There exists a smooth and proper DM-stack $\ov{\SS}_{g,n_{NS},n_R}$ over $\C$ representing the functor of families of stable supercurves of genus $g$ with
$n_{NS}$ NS punctures and $n_R$ Ramond punctures.}

\medskip

After we finished this work we learned about the work of Moosavian and Zhou \cite{MZ}, where Theorem A is also proved.
Their work also contains a lot of useful foundational results in algebraic supergeometry, e.g., proves the existence of Hilbert superschemes.
Another work with some foundational results, including the existence of Hilbert and Picard superschemes, is the forthcoming paper \cite{Hilbert-schemes}.

We mostly follow \cite{Deligne} in the part concerning deformation theory (extending it to the case of stable supercurves with punctures).
In the rest of the proof we avoid using the Artin's criterion and give a more direct proof based on the existing solution of the corresponding purely even moduli problem
due to Cornalba and Jarvis. Namely, the functor of families of stable supercurves
restricted to even bases is precisely the functor of generalized spin structures considered in \cite{Cornalba}, \cite{Jarvis}.

In the course of proof of Theorem A we find a natural relatively ample line bundle on any family of stable supercurves with punctures (see Theorem \ref{ample-thm}).
Although this is not strictly necessary for Theorem A (where one could use an algebraic space to replace the Hilbert scheme), this can be viewed as a super-analog
of the well known fact that the relative log canonical bundle is relatively ample for a family of stable curves.
The important difference is that in the case of a family of stable supercurves $X\to S$ the relative canonical bundle on the smooth part does not extend to a line bundle on the entire
family (the extension is not locally free at NS nodes). However, we prove that its square does extend to a line bundle which we still denote as $\om_{X/S}^2$ (one needs
a technical assumption that the corresponding map from $S$ to the moduli stack is smooth).
This line bundle plays an important technical role in the rest of the paper. In the hindsight, this is not too surprizing since the restriction of $\om_{X/S}^2$ to the reduced part
of moduli of supercurves recovers the relative dualizing sheaf of the corresponding usual family of stable curves.

One point which is not highlighted in \cite{Deligne} is that there is a canonical effective Cartier divisor $\De$ supported on the boundary of the compactification $\ov{\SS}_g$
(in the presence of odd variables, which are nilpotent, such a structure is not unique).
Essentially this structure is already seen from the study of deformations of the nodes of supercurves which can be of two types, NS and Ramond.
We also define a decomposition $\De=\De_{NS}+\De_R$ corresponding to these two types of nodes
and find two global expressions for the line bundle $\OO(\De)$ using Berezinians of certain natural morphisms of sheaves associated with the universal curve
(see Sec.\ \ref{boundary-str-sec}).

The second goal of this work is to study the canonical line bundle $K_{\ov{\SS}_g}$
over $\ov{\SS}_g$, i.e., the Berezinian of the cotangent vector bundle.
The super analog of Mumford isomorphism for smooth supercurves,
expressing the canonical line bundle of $\SS_g$ in terms of natural Berezinian line bundles was considered in
\cite{Voronov}, \cite{RSV} (see also \cite{Diroff} where some work is done in the punctured case).
Namely, it states that
$$K_{\SS_g}\simeq\Ber_1^5,$$
where
$$\Ber_1:= \Ber(R\pi_*\OO_X)\simeq \Ber(R\pi_*\om_{X/\ov{\SS}_g}),$$
where $\pi:X\to \SS_g$ is the
universal stable supercurve.

In the case of the moduli of stable supercurves (and in the presence of punctures)
we still have the line bundle $\Ber_1$ defined as above, however
the expression for the canonical bundle has to be corrected.

\bigskip

\noindent
{\bf Theorem B}. {\it Let $\ov{\SS}= \ov{\SS}_{g,n_{NS},n_R}$.
There exists a canonical isomorphism
$$K_{\ov{\SS}}\rTo{SM_{\ov{\SS}}}
\Ber_1^5\otimes \bigotimes_{i=1}^{n_{NS}}\Psi_i(-2\De_{NS}-\De_R),$$
where $\De_{NS}$ and $\De_R$ are the components of the boundary divisor corresponding to nodes of NS and Ramond type,
and $\Psi_i$ is the line bundle associated with the $i$th NS puncture $P_i$:
$$\Psi_i:=P_i^*\om_{X/\ov{\SS}}.$$

}

\medskip

In the case when there are no punctures the isomorphism becomes
$$K_{\ov{\SS}_g}\simeq \Ber_1^5(-2\De_{NS}-\De_R).$$

Finally, we study the restriction of the super-Mumford isomorphism $SM_{\ov{\SS}_g}$ to the NS boundary divisor, and show that it is related to the similar isomorphisms for lower genus. As in the classical case, each component of $\De_{NS}$ is an image of a natural gluing map $B\to \ov{\SS}_g$,
where $B$ is some lower genus moduli space with more punctures or a product of two such moduli spaces (see Sec.\ \ref{NS-boundary-gluing-sec}).

\bigskip

\noindent
{\bf Theorem C}. {\it Let $B\to \ov{\SS}_g$ be the gluing map to one of the components of $\De_{NS}$ from a lower genus (uncompactified) moduli space $B$.
There exists a natural isomorphism (see below) of the normal bundle
$$N_B:=\OO(\De)|_B\simeq \Psi_1^{-1}\ot \Psi_2^{-1},$$
such that the following diagram is commutative up to a sign
\begin{diagram}
K_{\ov{\SS}_g}(2\De)|_B&\rTo{SM_{\ov{\SS}_g}|_B}& \Ber_1^5|_B\\
\dTo{}&&\dTo{}\\
K_B\ot N_B&\rTo{SM_B}&(\Ber_1^B)^5
\end{diagram}
}

The identification of $N_B$ in Theorem C comes from the identification of $\De$ as the vanishing
locus of a morphism induced on Berezinians by the morphism
$$R\pi_*(\Om_{X/\ov{\SS}_g})\to R\pi_*j_*(\Om_{U/\ov{\SS}_g}),$$
where $j:U\to X$ is the embedding of the smooth locus (see Sec.\ \eqref{boundary-equation-sec}).

The components of the Ramond boundary divisor $\De_R$ have a more complicated relation to lower genus moduli spaces (see Sec.\
\ref{R-boundary-gluing-sec}), and in this case we formulate a conjectural analog of Theorem C in Sec.\ \ref{R-splitting-sec}.

The paper is organized as follows.
In Section \ref{basics-sec} we review the basics on stable supercurves starting from the definitions.
In particular, in \ref{local-descr-sec} we discuss standard local coordinates of smooth supercurves (near an ordinary point and near a Ramond
puncture).

Then in \ref{NS-correspondence-sec} we discuss a well-known correspondence
between the NS-punctures and divisors.

In Section \ref{infinit-sec} we discuss some infinitesimal results on supercurves. We recall the construction from \cite{Deligne} of miniversal deformation families
of two types of node singularities of stable supercurves and consider sheaves of infinitesimal automorphisms of stable supercurves (possibly with punctures).

In Section \ref{def-sec} we study the deformation functor of a stable supercurve $X_0$ with punctures. The main result is that this functor is smooth (more precisely,
it is smooth over the product of deformation functors of the singularities of $X_0$). We first study local deformations: of affine neighborhoods of a smooth point,
of a Ramond puncture, and of a singular point (see \ref{no-R-punct-def-sec}, \ref{R-punct-def-sec} and Lemma \ref{aff-formal-node-smooth-lem}).
Then we prove the smoothness result for global deformations, Proposition \ref{global-def-prop}.

In Section \ref{square-sec} we prove that an extension of the square of the relative canonical bundle $\om_{U/S}^2$ from the smooth locus $U\sub X$
of a sufficiently nice family $X\to S$ of stable supercurves is a line bundle, and that after some corrections at the punctures, it becomes relatively ample
(see Theorems \ref{can-square-thm} and \ref{ample-thm}).
We then show that the inverse of this line bundle shows up in the computation of the sheaf of infinitesimal automorphisms (see Theorem \ref{inf-aut-thm}).

In Section \ref{proof-thm} we prove Theorem A. Mostly we use the previous results on deformation theory and the known results on the moduli of
generalized spin curves.

In Section \ref{KS-map-sec} we study the behavior of the Kodaira-Spencer map for a family of smooth supercurves degenerating to a stable supercurve.
We consider the classical case of (even) curves in \ref{KS-class-sec}, then the case of an NS node in \ref{super-KS-NS-sec} and the case of a Ramond node
in \ref{super-KS-R-sec}. The main observation is that the Kodaira-Spencer map has a natural extension over the entire base involving
the subsheaf of the tangent space to the base consisting of vector fields preserving the degeneration divisor.

In Section \ref{boundary-str-sec} we study the boundary divisor $\De$ of the moduli of stable supercurves. We give a definition of the boundary divisor as a Cartier divisor
(which is not automatic since we work with nonreduced spaces). We compute the corresponding line bundle in terms of the complex $[\Om_{X/S}\to j_*\Om_{U/S}]$, where $U\sub X$ is the smooth locus of a family $X\to S$ (see \ref{boundary-equation-sec}). We prove
in \ref{normal-cross-sec} that $\De$ is a normal crossing divisor and define in \ref{De-NS-R-sec}
the subdivisors $\De_{NS}$ and $\De_R$ corresponding to the NS and Ramond type nodes such that $\De=\De_{NS}+\De_R$.
Then we discuss in \ref{NS-boundary-gluing-sec} and
\ref{R-boundary-gluing-sec} the gluing maps from lower genus moduli spaces to the boundary components.
Note that in the case of a Ramond node there is an extra odd parameter involved in the gluing.

In Section \ref{canonical-bundle-sec} we study the canonical line bundle over the moduli of stable supercurves with the goal of proving Theorem B.
The identification of the canonical line bundle for the moduli of smooth supercurves is a combination of the isomorphism coming from the Kodaira-Spencer map and of an analog of
the Mumford's isomorphism between different Berezinian line bundles. We have to investigate what happens with both these ingredients near the boundary divisor.
For the Kodaira-Spencer map this was done in Section \ref{KS-map-sec}.
The main new nontrivial computation is that of the behavior of the super Mumford's isomorphism as the supercurve degenerates (see \ref{super-Mum-NS-sec}).
Extending this picture to supercurves with punctures is relatively easy and is done in \ref{can-line-bun-punctures-sec}.

In Section \ref{Splitting-sec} we prove Theorem C. Again the bulk of the argument is the study of the restriction of super Mumford's isomorphism to the boundary divisor
in \ref{NS-splitting-sec}. Then in \ref{KS-splitting-sec} we do the same for the isomorphism coming from the Kodaira-Spencer map. In \ref{R-splitting-sec}
we describe a conjectural picture for the case of a Ramond boundary component.

In Appendix \ref{ample-sec} we prove a relative ampleness criterion for flat morphisms of superschemes (see Proposition \ref{ample-crit-prop}), which we
use to prove Theorem \ref{ample-thm}.

\subsection*{Conventions}
All the rings are assumed to be $\Z_2$-graded supercommutative with $1$.
We say that a ring (resp., a superscheme) is {\it even} if its odd component (resp., the odd component of the structure sheaf) is zero.
We often say ``subscheme" for brevity where we should say ``sub-superscheme".
For a sheaf $F$ of $\OO_X$-modules on a superscheme $X$ we denote by $F^+$ and $F^-$ its even and odd parts.
On any superscheme $X$ we denote by $\NN_X\sub \OO_X$ the ideal generated by odd functions.
We denote by $X_{\bos}$ the usual scheme with the same underlying topological space as $X$ and with the structure sheaf
$\OO_X/\NN_X$.

\subsection*{Acknowledgements}
We thank Ugo Bruzzo and Daniel Hern\'andez Ruip\'erez for useful discussions and

for telling us about the work \cite{MZ}.
G.F. is partially supported by the National Centre of
Competence in Research ``SwissMAP --- The Mathematics of Physics'' of the
Swiss National Science Foundation.

D.K. is partially supported by the ERC under grant agreement 669655.
A.P. is partially supported by the NSF grant DMS-2001224,

and within the framework of the HSE University Basic Research Program and by the Russian Academic Excellence Project `5-100'.

\section{Stable supercurves}\label{basics-sec}

In this section we discuss some basic facts about stable supercurves and their families, starting with definitions. In particular,
we discuss local descriptions of nodes and punctures, the correspondence between the NS punctures and divisors, and the
connection with generalized spin structures.

\subsection{Superstacks}

We refer to \cite{CCF} and \cite[Sec.\ 2]{BHR} for some basics on superschemes.
The notion of an algebraic (super)stack is developed similarly to the classical case (see \cite[Def.\ (5.1)]{Artin}):
one considers a category fibered in groupoids $\XX$ over the category of Noetherian superalgebras over $\C$,
which is a limit preserving stack (see \cite[(1.1)]{Artin}), such that the diagonal $\XX\to \XX\times \XX$ is
representable and there exists a smooth surjective morphism from a scheme to $\XX$ (see \cite[Sec.\ 3]{CV}).
If the latter morphism can be chosen to be \'etale, then one gets the notion of Deligne-Mumford (DM) stack.

For example, it is proved in \cite{CV} that the moduli stack of (smooth) supercurves is a smooth separated DM stack over $\C$.

\subsection{Definition of families of stable supercurves}\label{def-stable-supercurve-sec}

\begin{definition}
Let $S$ be a superscheme. A family of {\it smooth supercurves with Ramond punctures} is a proper smooth
superscheme $X$ over $S$ of relative dimension $1|1$, equipped with a subbundle $\DD\sub \TT_{X/S}$ of rank $0|1$,
such that the map given by the commutator of vector fields
$$\DD^{\ot 2}\to \TT_{X/S}/\DD$$
(which is a map of line bundles of rank $1|0$) is injective and its divisor of vanishing is the disjoint union
$$R=\sqcup_{i=1}^{n_R} R_i,$$
where each $R_i$ is smooth of dimension $0|1$ over $S$.
\end{definition}

In the above situation the map $D_1\ot D_2\mapsto \frac{1}{2}[D_1,D_2]$ induces an isomorphism
\begin{equation}\label{Ramond-D2-isom}
\DD^{\ot 2}\rTo{\sim} \TT_{X/S}/\DD(-R).
\end{equation}
Thus, we have an exact sequence
$$0\to \DD\to \TT_{X/S}\to \DD^{\ot 2}(R)\to 0,$$
or dually,
$$0\to \DD^{-2}(-R)\to \Om_{X/S}\to \DD^{-1}\to 0.$$
From this we get an isomorphism $\om_{X/S}=\Ber(\Om_{X/S})\simeq \DD^{-1}(-R)$.
Hence, we get a surjective morphism
$$\de:\Om_{X/S}\to \DD^{-1}\simeq \om_{X/S}(R),$$
or equivalently, a derivation $\OO_X\to \om_{X/S}(R)$, trivial on $\OO_S$.
Note that $\DD$ is recovered from $\de$ as the orthogonal to $\ker(\de)$.

\begin{example}\label{de-comp-ex}
Suppose $X$ is an open subscheme in $\A^{1|1}_S$ with relative coordinates $(z,\th)$ and
$\DD$ is generated by $D=\pa_\th+f\th\pa_z$, for some even function $f$ on $X$.
Then
$$\frac{1}{2}[D,D]=f\pa_z,$$
so the condition that $\DD$ defines a structure of a supercurve on $X$ (with no punctures)
is that $f$ is invertible.
Thus, the canonical isomorphism
$$\DD^{\ot 2}\to \TT_{X/S}/\DD$$
sends $D\ot D$ to $f\pa_z \mod\DD$.
Hence, the isomorphism
$$\DD\rTo{\sim}\Ber(\TT_{X/S})\simeq \om_{X/S}^{-1}$$
sends $D$ to $f[dz|d\th]^{-1}$.
Thus, we can compute the dual of the embedding $\DD\to \TT_{X/S},$
$$\de:\Om_{X/S}\to \DD^{-1}\simeq \om_{X/S}.$$
Namely, $\de$ sends $dz$ to $\th[dz|d\th]$ and $d\th$ to $f^{-1}[dz|d\th]$. Thus, if we view $\de$ as a derivation,
we have
$$\de(\phi)=D(\phi)\cdot f^{-1}[dz|d\th].$$
\end{example}

\begin{definition}
A family of {\it stable (resp., prestable) supercurves with punctures} over $S$, is a superscheme $X/S$,
proper, flat and relatively Cohen-Macaulay, together with
\begin{itemize}
\item
a collection of disjoint closed subschemes called {\it NS-punctures} and {\it R-punctures}
$$P_i\sub X, \ i=1,\ldots,n_{NS}; \ \ R_i\sub X, i=1,\ldots,n_R,$$
such that the projection $P_i\to S$ is an isomorphism, and each $R_i$ is a Cartier divisor;
\item
a derivation
$$\de:\OO_X\to \om_{X/S}(R),$$
trivial on pull-backs of functions on $S$, where $R=\sum R_i$. Here $\om_{X/S}$ is the relative dualizing sheaf
so that $\pi^!\OO_S=\om_{X/S}[1]$ (the fact that it is a sheaf follows from the Cohen-Macaulay property).
\end{itemize}
We impose the following additional properties.
\begin{itemize}
\item There is an open fiberwise dense subset $U\sub X$ such that $U/S$ is smooth of dimension $1|1$,
and $P_i\sub U$, $R_i\sub U$;
\item the derivation $\de$ corresponds to a structure of a smooth supercurve with Ramond punctures on $(U,(R_i))$ over $S$;
\item $(X^{\red}, (P_i^{\red}), (R_i^{\red}))$ is a family of usual stable (resp., nodal) pointed curves over $S^{\red}$;
\item on every geometric fiber $X_s$, $\de^-$ induces an isomorphism
$$\OO_{X_s}^-\rTo{\sim}\om_{X_s}^-(\sum R_{is}).$$
\end{itemize}
We say that an isomorphism of stable supercurves over $S$ is {\it superconformal} if it is compatible with derivations $\de$
We say that a vector field on $X$ is {\it superconformal} if it preserves $\de$.
\end{definition}

As observed in [Deligne], if $U\sub X$ is the maximal open subset which is smooth over $S$, then
$\de$ induces a structure of a smooth supercurve on $U\setminus \cup_i R_i$ over $S$.

\begin{remark}\label{even-case-rem}
Assume that $S$ is even. Then $X=(C,\OO_C\oplus L)$, where $C/S$ is a family of prestable curves, $L=\OO_X^-$ is
a coherent sheaf on $C$, flat and relatively Cohen-Macaulay over $S$, $L$ is locally free of rank $1$ over a fiberwise
dense open in $C$. The marked points $P_i$ and $R_i$ correspond to marked points $p_i:S\to C$ and $r_i:S\to C$.
Namely, $P_i$ is the image of the composition $S\rTo{p_i}C\to X$, while $R_i$ is the preimage of $r_i(S)$ under the projection
$X\to C$.
Furthermore, $\de^-$ gives an isomorphism
\begin{equation}\label{gen-spin-str-eq}
L\rTo{\sim}\und{\Hom}(L,\om_C(\sum r_i)).
\end{equation}
\end{remark}

\begin{lemma}\label{isom-stable-lem}
Let $(X,P_\bullet,R_\bullet)$ and $(X',P'_\bullet,R'_\bullet$) be a pair of prestable supercurves over $S$,
and let $\varphi:X\rTo{\sim} X'$ be an isomorphism of superschemes over $S$, sending $P_i$ to $P'_i$ and
$R_j$ to $R'_j$. Assume that the restriction of $\varphi$ to a fiberwise dense open subscheme $U\sub X$, which is smooth over $S$, is an isomorphism of smooth
supercurves. Then $\varphi$ is an isomorphism of stable supercurves.
\end{lemma}

\begin{proof}
Let $j:U\to X$ be the inclusion.
By assumption, we have $\varphi^*\om_{X'/S}(R')\simeq \om_{X/S}(R)$.
Thus, we can view $\varphi^*\de_{X'}$ as a derivation $\OO_X\to \om_{X/S}(R)$, and we have
to prove the equality $\varphi^*\de_{X'}=\de_X$. Since we know that this equality holds over $U$,
it is enough to check that the map
$$\om_{X/S}(R)\to j_*j^*\om_{X/S}(R)$$
is injective. But this follows from Cohen-Macaulay property: otherwise in some neighborhood of a point in $X$ the kernel
would be a subsheaf with finite support over $S$. So we would get a nonzero morphism
$$\OO_Z\to \om_{X/S}=\pi^!\OO_S[-1],$$
with $Z$ finite over $S$. By duality it would correspond to a nonzero morphism $\pi_*\OO_Z\to \OO_S[-1]$
which is impossible.

\end{proof}

\subsection{Local descriptions}\label{local-descr-sec}

It is well known (see e.g., \cite[Lem.\ 1.2]{LR} or \cite[Lem.\ 3.1]{DW} in the absolute case)
that locally in classical topology near a point of a smooth supercurve, there exist relative coordinates $(z,\th)$ such that
$\DD$ is generated by $D=\partial_\th+\th\partial_z$ and $\de$ is given by $\de(f)=D(f)\cdot [dz|d\th]$.
We show that the same assertion holds with respect to the \'etale topology and also consider an analogous statement for the case of Ramond punctures
(cf.\ \cite[Prop.\ 3.6]{BHR}).

\begin{lemma}\label{local-descr-smooth-lem}
(i) Let $X/S$ be a smooth supercurve and let $(z,\th)$ be a pair of even and odd local functions such that $dw$ and $d\th$ generate the relative cotangent bundle near a point $p\in X$.
Then there exists another pair $(w,\eta)$ like this defined in an \'etale neighborhood of $p$, with $w\equiv z\mod\NN_X$, such that $\DD$ is generated
by $\partial_\eta+\eta\partial_w$.

\noindent
(ii) Now let $X/S$ be a smooth supercurve with a Ramond puncture $R\sub X$, and let $(z,\th)$ be a pair of even and odd local functions such that $dw$ and $d\th$ generate the relative cotangent bundle and such that the ideal of $R$ is generated by $z$.
Then locally in \'etale topology there exist a change of coordinates to $(w,\eta)$, with $w\equiv z\mod\NN_X$,
such that $\DD$ is generated by $\partial_\eta+\eta w\partial_w$.
\end{lemma}

\begin{proof}
(i) Let $\DD$ be generated by an odd vector field $D$ of the form $D=f\partial_\th+g\partial_z$. One has
$$\frac{1}{2}[D,D]\equiv f\partial_\th(g)\cdot \partial_z \mod\NN_X\TT_{X/S}.$$
Note also that modulo nilpotents $D$ reduces to $f\partial_\th$.
Hence, in order for $D$ and $[D,D]$ to generate $\TT_{X/S}$ both $f$ and $\partial_\th(g)$ should be invertible.

Thus, we can assume that $\DD$ is generated by some vector field of the form $D=\partial_\th+g\partial_z$, where $g=g_1(z)+g_0(z)\th$,
with $g_0$ even and $g_1$ odd.
Furthermore, $g_0$ is invertible.
Let us look for $w$ and $\eta$ in the form
\begin{align*}
&\eta=a_0(z)\cdot \th,\\
&w=z+a_1(z)\th,
\end{align*}
where $a_0$ is invertible even and $a_1$ is odd.
Changing to the new coordinates we get
$$D=[a_0+ga'_0\th]\partial_\eta+[-a_1+g(1+a'_1\th)]\partial_w.$$
In order for $\DD$ to be generated by $\partial_\eta+\eta\partial_w$ we need
the equation
$$-a_1+g(1+a'_1\th)=(a_0+ga'_0\th)\cdot a_0\th$$
to be satisfied. This is equivalent to the system
\begin{align*}
&-a_1+g_1=0,\\
&g_0+g_1a'_1=a_0^2.
\end{align*}
Since $g_0$ is invertible, in an \'etale neighborhood we can choose $a_0$ such that $a_0^2=g_0+g_1g'_1$.
This $a_0$ together with $a_1=g_1$ is a solution.

\medskip

\noindent
(ii) Let $D=f\partial_\th+g\partial_z$ be a generator of $\DD$. From the calculation in the beginning of part (i), we get an isomorphism
$$(\TT_{X/S}/\DD+[\DD,\DD])\ot \OO_{X_\bos}\simeq \OO_{X_\bos}/(f\partial_\th(g))\cdot \partial_z\oplus \OO_{X_\bos}/(f)\cdot \partial_\th$$
where $\OO_{X_\bos}=\OO_X/\NN_X$.
Thus, this sheaf surjects onto $\OO_{X_\bos}/(f)\oplus \OO_{X_\bos}/(f)$.
Since this quotient has to be isomorphic to $\OO_X/(\NN_X+(z))$, we deduce that $f$ is invertible modulo $\NN$, so $f$ is invertible.

Thus, we can assume that $\DD$ is generated by $D=\partial_\th+g\partial_z$, where
$g=g_1(z)+g_0(z)\th$. Then we have
$$\frac{1}{2}[D,D]=[g_0+g_1g'_1+(g_1g'_0-g_0g'_1)\th]\cdot \partial_z,$$
where $f'$ denotes the derivative with respect to $z$.
In order for the commutator map $\DD^{\ot 2}\to \TT_{X/S}/\DD$ to vanish exactly on the divisor $(z)$ we should have
$$g_0+g_1g'_1+(g_1g'_0-g_0g'_1)\th=z(u_0(z)+u_1(z)\th),$$
with $u_0$ invertible. In other words,
\begin{equation}\label{g0-g1-eq}
\begin{array}{l}
g_0+g_1g'_1=zu_0, \\
g_1g'_0-g_0g'_1=zu_1.
\end{array}
\end{equation}
We claim that this implies that $g_1$ is divisible by $z$. Indeed, let us write
$$g_0\equiv p_0+q_0z \mod (z^2),$$
$$g_1\equiv p_1+q_1z\mod (z^2),$$
where $p_i$ and $q_i$ are functions on the base. Note that since $g_1$ is odd, from
the first of the equations \eqref{g0-g1-eq} we get that $q_0$ is invertible. Also, looking at the constant terms of these equations we get
\begin{align*}
&p_0+p_1q_1=0,\\
&p_1q_0-p_0q_1=0.
\end{align*}
This implies that
$$p_1(q_0+q_1^2)=p_1q_0=0.$$
Since $q_0$ is invertible, we deduce that $p_1=0$ which proves our claim that $g_1$ is divisible by $z$.

Now we are going to make the same change of coordinates as in (i) with an additional constraint that $a_1$ is divisible by $z$.
Since $\eta w=a_0z\th$, we have to solve the equation
$$-a_1+g(1+a'_1\th)=(a_0+ga'_0\th)\cdot a_0z\th,$$
or equivalently, the system
\begin{align*}
&-a_1+g_1=0,\\
&g_0+g_1a'_1=a_0^2z.
\end{align*}
Thus, we get the solution by taking $a_0$ to be the square root of $u_0$
and $a_1=g_1$ (which is divisible by $z$).
\end{proof}

\begin{definition} We refer to $(w,\eta)$ as in Lemma \ref{local-descr-smooth-lem} as {\it standard coordinates}.
\end{definition}

By Example \ref{de-comp-ex}, if $(z,\th)$ are standard coordinates on a smooth supercurve (away from punctures)
then the canonical derivation is given by
$$\de(f)=(\partial_\th+\th \partial_z)(f)\cdot [dz|d\th].$$
On the other hand, if $(z,\th)$ are standard coordinates near a Ramond puncture then
$$\de(f)=(\partial_\th+\th z\partial_z)(f)\cdot [\frac{dz}{z}|d\th].$$

In the rest of this subsection we consider only the absolute case, i.e., $S=\Spec(\C)$.
Let $X$ be a stable supercurve over $\C$. We denote by $(C,L)$ be the underlying generalized spin curve, and fix a node
$q\in C$. Recall that by definition $X$ is required to be Cohen-Macaulay (CM).
Using the classification of CM sheaves of rank $1$ on the nodal singularity
(see e.g. \cite[Sec.\ 1]{Jarvis-95}), we obtain that there are two possibilities for $L$:

\begin{itemize}
\item $L$ is locally free near $q$. Then we say that this is a {\it Ramond node} (or {\it R-node}).

\item $L$ is the push forward of a line bundle on the normalization of $C$ near $q$. Then we say that this is a {\it NS node}.
\end{itemize}

Furthermore, for a stable curve over $\C$ we have the following local descriptions near the nodes (see \cite{Deligne}).

\begin{itemize}
\item Near a Ramond node $X$ has coordinates $(z_1,z_2,\th)$ (where $\th$ is odd) subject to $z_1z_2=0$. The complement of the node is the union of two branches
$U_1$ and
$U_2$, where $z_i$ is invertible on $U_i$, and $\om_X$ is free with the basis $\bb$ given by
$$\bb=\begin{cases} [\frac{dz_1}{z_1}|d\th] & \text{ on } U_1,\\ -[\frac{dz_2}{z_2}|d\th] & \text{ on } U_2.\end{cases}$$
The derivation $\de$ is given by
$$\de(f)=\begin{cases} (\pa_\th+\th z_1\pa_{z_1})(f)\cdot \bb & \text{ on } U_1,\\ (\pa_\th-\th z_2\pa_{z_2})(f)\cdot \bb & \text{ on } U_2.\end{cases}$$

\item Near an NS node $X$ has coordinates $z_1,z_2,\th_1,\th_2$ subject to the equations
$$z_1z_2=z_1\th_2=z_2\th_1=\th_1\th_2=0.$$
The complement of the node is given again as the union of two branches $U_1$ and $U_2$, where $z_i$ is invertible on $U_i$
and $(z_i,\th_i)$ form coordinates on $U_i$. The generators $[dz_1|d\th_1]$ and $[dz_2|d\th_2]$ of $\om_{U_1}$ and of $\om_{U_2}$ extend to sections
of $\om_X$ (zero on another branch), however, they do not generate it: there is an extra section
$$s_0=\begin{cases} \frac{\th_1}{z_1}[dz_1|d\th_1] & \text{ on } U_1,\\ -\frac{\th_2}{z_2}[dz_2|d\th_2] & \text{ on } U_2.\end{cases}.$$
The derivation $\de$ is given by
$$\de(f)=\begin{cases} (\pa_{\th_1}+\th_1 \pa_{z_1})(f)\cdot \bb & \text{ on } U_1,\\ (\pa_{\th_2}+\th_2\pa_{z_2})(f)\cdot \bb & \text{ on } U_2.\end{cases}.$$
\end{itemize}

\subsection{Correspondence between NS-punctures and divisors}\label{NS-correspondence-sec}

Here we recall a natural correspondence between NS-punctures $P\sub X$ and effective Cartier divisors $D$ supported at $P$, for a smooth supercurve $X/S$
(with arbitrary base $S$).

It is based on the fundamental fact that for a smooth supercurve $X/S$, the composed map
\begin{equation}\label{smooth-superconf-map}
\TT^{sc}_{X/S}\to \TT_{X/S}\to \TT_{X/S}/\DD\simeq \DD^{\ot 2}\simeq \om_{X/S}^{-2}
\end{equation}
is an isomorphism of sheaves, where $\TT^{sc}_{X/S}\sub \TT_{X/S}$ is the subsheaf of superconformal vector fields (those preserving $\DD$).
Indeed, the proof is easily obtained from the existence of standard coordinates as in Lemma \ref{local-descr-smooth-lem}
(see \cite[Lem.\ 2.1]{LR}). We denote by
$$\a:\om_{X/S}^{-2}\rTo{\sim} \TT^{sc}_{X/S}$$
the inverse isomorphism.

\begin{definition} We say that an effective Cartier divisor $D\sub X$ corresponds to an NS-puncture $P\sub X$ if $D$ is supported at $p$
and the following property holds. If the ideal of $D$ is generated
locally by a function $f$ then the ideal of $P_i$ is generated by $(f,A(f))$, where $A$ is any generator of the distribution
giving the superconformal structure.
\end{definition}

The fact that in this way we get a bijective correspondence is a part of the next

\begin{lemma}\label{NS-div-corr-lem}
For a smooth supercurve $X/S$ and an NS-puncture $P\sub X$ there exists a unique effective divisor $D$ (locally generated by an even non-zero-divisor),
such that the following square is commutative
\begin{diagram}
\om_{X/S}^{-2}&\rTo{\a}& \TT^{sc}_{X/S}\\
\dTo{}&&\dTo{}\\
\om_{X/S}^{-2}|_{D}&\rTo{\b}&\TT_{X/S}|_{P}
\end{diagram}

where $\b$ is an isomorphism of sheaves of $\OO_S$-modules.
Furthermore, the ideal of $P$ is recovered from a local equation $f$ of $D$ as $(f,A(f))$. In particular,
we have inclusion of subschemes $P\sub D$.
\end{lemma}

\begin{proof} The isomorphism $\a$ locally has form
$$f\cdot \bb^{-2}\mapsto f\cdot \partial_z+\frac{1}{2}(-1)^{|f|}A(f)\cdot A,$$
where $A=\pa_{\th}+\th\pa_z$
and $\bb=[dz|d\th]$ is a generator of $\om_X$ (see the proof of \cite[Lem.\ 2.1]{LR}, where however the factor $\frac{1}{2}$ is missing).

Let $\a_{P}$ be the composition of $\a$ with the projection to $\TT_{X/S}|_{P}$.
Locally the ideal of $P$ has form $(z,\th-a)$, for some odd function $a$ on the base.
Then it is easy to see that
$$\a_{P}((z+a\th)f\cdot\bb^{-2})=0,$$
while $\a_{P}(1)$ and $\a_{P}(\th)$ form a basis of $\TT_{X/S}|_{P}$.
This shows that the assertion holds for $D=(z+a\th)$.
\end{proof}

Note that since the composition $P\hra D\to S$ is an isomorphism, we have a canonical splitting
\begin{equation}\label{NS-div-splitting-eq}
\OO_D\simeq \OO_S\oplus I_P/I_D.
\end{equation}

\begin{cor}\label{NS-puncture-Tsc-isom-cor}
The map \eqref{smooth-superconf-map}
induces an isomorphism of sheaves
$$\TT^{sc}_{(X,P)/S}\rTo{\sim} \om_{X/S}^{-2}(-D),$$
where $\TT^{sc}_{(X,P)/S}\sub \TT^{sc}_{X/S}$ is the sheaf of superconformal vector fields preserving the ideal of $P$.
\end{cor}

\subsection{Supercurves over even bases and spin curves}\label{even-moduli-sec}

If we only consider even bases, the functor of (stable) supercurves with punctures coincides with the functor
of stable curves with punctures equipped with generalized spin structures
i.e., torsion free sheaves $L$ of rank $1$ equipped with an isomorphism \eqref{gen-spin-str-eq}
(see Remark \ref{even-case-rem}).

The moduli stack $\ov{\SS}_{g,n_{NS},n_R}$ of stable spin curves of genus $g$
with $n_{NS}$ NS-punctures and $n_R$ Ramond punctures
was studied in \cite{Jarvis}, \cite{AJ}, \cite{JKV}, as a particular case of the stack of stable $r$-spin curves.
In particular, it shown to be a smooth proper Deligne-Mumford stack.
Note that although for $r>2$ there are different versions of the functor of stable $r$-spin curves, however, in the case $r=2$, they all coincide (see
\cite{Jarvis-95}).

Let us recall how the generalized spin structures over stable curves look like.
A generalized spin structure $L$ over a smooth curve $C$ with punctures is a line bundle $L$ equipped with an isomorphism
$$L^2\simeq \om_C(r_1+\ldots+r_n),$$
where $(r_\bullet)$ are the Ramond punctures.

Now let $(C,p_\bullet,r_\bullet)$ be a stable curve, and
let $\rho:\wt{C}\to C$ be the partial normalization map, resolving a single node $q\in C$.
Let us equip $\wt{C}$ with punctures in the following way:
first, it inherits all the punctures (NS and Ramond) of $C$. Secondly,
we mark the two points in $\rho^{-1}(q)$ as two NS (resp., Ramond) punctures on $\wt{C}$ if $q$ is a NS (resp., Ramond)
node on $C$.

For a generalized spin structure $L$ over $C$
let us define the sheaf $\wt{L}$ on $\wt{C}$ as the quotient of $\rho^*L$ by the torsion subsheaf.
Then $\wt{L}$ is a line bundle on $\wt{C}$ and
$\wt{L}$ is a generalized spin structure on $\wt{C}$ with NS and Ramond punctures defined as above.

More precisely, if $q$ is a Ramond node then $L$ is locally free near $q$ and $\wt{L}=\rho^*L$ (no torsion subsheaf appears in this case).
If $q$ is an NS node then $L$ is locally isomorphic to the ideal of the node, whereas $\wt{L}$ is locally free near the preimage of $q$.
In the latter case we have a natural isomorphism $L\simeq \rho_*\wt{L}$.

\section{Infinitesimal study}\label{infinit-sec}

In this section, following \cite{Deligne}, we describe miniversal deformations of both types of nodal singularities of stable supercurves,
and show how this leads to the proof of absence of infinitesimal automorphisms. We also study the sheaf of infinitesimal automorphisms
in the presence of punctures.

\subsection{Miniversal deformations of the nodes}\label{nodes-def-sec}

These constructions are from \cite{Deligne}. They are miniversal deformations of two types of nodal singularities of supercurves.

In both cases the base $S$ of deformations is $\Spec(k[t])$, where $t$ is even.

\subsubsection{Ramond node}\label{univ-def-node-I}

Define $X/S$ as a subscheme of the affine space $S\times \A^{2|1}$ over $S$ with coordinates $z_1,z_2,\th$, given
by the equation
$$z_1z_2=t.$$
Note that we have a natural trivialization $\bb$ of $\om_{X/S}$.

Over the open subset where $z_1\neq 0$ (resp., $z_2\neq 0$), we have $\bb=[\frac{dz_1}{z_1}|d\th]$
(resp., $\bb=[-\frac{dz_2}{z_2}|d\th]$), and $\de$ is given by $(\pa_\th+\th z_1\pa_{z_1})\cdot\bb$
(resp., $(\pa_\th-\th z_2\pa_{z_2})\cdot\bb$).

\subsubsection{NS node}\label{univ-def-node-II}

Define $X/S$ as a subscheme of the affine space $S\times \A^{2|2}$ over $S$ with coordinates $z_1,z_2,\th_1,\th_2$, given
by the equations
$$z_1z_2=-t^2, \ \ z_1\th_2=t\th_1, \ \ z_2\th_1=-t\th_2, \ \ \th_1\th_2=0.$$

Over the chart where $z_i\neq 0$ ($i=1,2$),
\begin{equation}\label{NS-miniversal-de-formula}
\de(f)=(\pa_{\th_i}+\th_i\pa_{z_i})(f)[dz_i|d\th_i].
\end{equation}

The following result is proved in \cite{Deligne}.

\begin{theorem}\label{miniversal-Deligne-thm}
The above two families induce miniversal deformations of the completed rings of the two types of nodes.
\end{theorem}

\begin{proof}
The main calculation done in \cite{Deligne} is that of the tangent space to the deformation functor for the two types of singularities.
Namely in both cases, there are no odd infinitesimal deformations and there is a $1$-dimensional space of even deformation coming
from the above families over $\C[t]$. This easily implies that the there are no obstructions, so the deformation functor is smooth,
and the assertion follows.
\end{proof}

\subsection{Sheaf of infinitesimal automorphisms}\label{inf-aut-sh-sec}

For a moment let us consider only supercurves over the point base $\Spec(\C)$.

\begin{definition} We define the sheaf of infinitesimal automorphisms $\AA$ of any geometric structure of the form (superscheme over $\C$
plus extra structure) as follows: $\AA^+$ is given by automorphisms of the trivial family of these structures over $\C[\eps]/(\eps^2)$,
where $\eps$ is even, while $\AA^-$ is given by automorphisms of the trivial family over $\C[\tau]/(\tau^2)$, where
$\tau$ is odd. For example, the sheaf of infinitesimal automorphisms of a superscheme is exactly the tangent sheaf.
\end{definition}

For a stable supercurve $(X,P_\bullet,R_\bullet)$ over $\C$, we denote by
$$\AA_{X,P_\bullet,R_\bullet}\sub \TT_X$$
the sheaf of infinitesimal automorphisms.
Note that the space of locally trivial
infinitesimal deformations of $(X,P_\bullet,R_\bullet)$ is $H^1(X,\AA_{X,P_\bullet,R_\bullet})$.

On the smooth locus, away from punctures, $\AA_X\simeq \TT^{sc}_X$ projects isomorphically to $\TT_X/\DD$ (see \eqref{smooth-superconf-map}).
More precisely, for $\OO_X=\OO_C\oplus L$, we have
$$\AA_X\simeq \TT_C\oplus L\ot \TT_C.$$
Here $\AA_X^+=\TT_C$ acts naturally on $\OO_C$ and on $L$ by the formula
$$v\cdot \th=\frac{L_v(\th^2)}{2\th},$$
where $\th^2\in \om_C$ and $L_v$ is the Lie derivative.
On the other hand, the action of $\varphi\ot v\in L\ot \TT_C$ on $(f,\psi)\in \OO_C\oplus L$ is given by
$$(\varphi\ot v)\cdot (f,\psi)=(\lan\varphi\cdot\psi, v\ran,\lan v,df\ran\cdot \varphi)\in \OO_C\oplus L.$$

Note that if $j:U\hra X$ is a smooth locus, then $\AA_X$ is the subsheaf of $j_*\AA_U$ consisting of
those automorphisms over $U$ that send $\OO_X$ to $\OO_X$.

\begin{definition}
Now let us consider a stable supercurve with punctures
$(X,P_\bullet,R_\bullet)$ over any base $S$.
We define
$$\AA_{X/S}=\AA_{(X,P_\bullet,R_\bullet)/S}\sub \TT_{X/S}$$
as the subsheaf consisting of derivations $v$ in $\TT_{X/S}$ preserving the punctures and preserving the distribution $\DD$ over the smooth locus.
We still call $\AA_{X/S}$ the sheaf of infinitesimal automorphisms.
\end{definition}

In the case $S=\Spec(\C)$ this agrees with our previous definition. Indeed this is clear on the smooth locus $U\sub X$.
Then we use the property that $\AA_X$ is recovered as the intersection of $\TT_X$ with $j_*\AA_U$ in $j_*\TT_U$.
Note however that the formation of $\AA_{X/S}$ is not compatible with the base change in general.

In the smooth case we have the following useful identification of the sheaf of infinitesimal automorphisms over an arbtrirary (not necessarily even) base.

\begin{prop}\label{inf-aut-smooth-punct-prop}
Let $(X,P_\bullet,R_\bullet)\to S$ be a smooth supercurve with punctures, where $(P_i)_{i\in I}$ are NS punctures
and $(R_j)_{j\in J}$ are Ramond punctures.
Then one has a natural isomorphism
$$\AA_{(X,P_\bullet,R_\bullet)/S}\simeq \TT^{sc}_{(X,P_\bullet)/S}\simeq \om_{X/S}^{-2}(-\sum_{i\in I} D_i-2\sum_{j\in J} R_j),$$
where $D_i\sub X$ is a divisor associated with the NS puncture $P_i$ (see Sec.\ \ref{NS-correspondence-sec}).
\end{prop}

\begin{proof}
The first isomorphism corresponds to the fact that the Ramond punctures are recovered from the distribution $\DD\sub \TT_{X/S}$ as the vanishing divisor
of the $\OO_X$-linear map
$$\DD^{\ot 2}\to \TT_{X/S}/\DD$$
induced by the Lie bracket.

Next, we observe that there is a natural isomorphism
$$\TT_{X/S}/\DD\simeq \DD^2(\sum_j R_j)\simeq \om_{X/S}^{-2}(-\sum_j R_j),$$
so we can consider the composed map
$$\TT^{sc}_{(X,P_\bullet)/S}\to \TT_{X/S}/\DD\simeq  \om_{X/S}^{-2}(-\sum_j R_j).$$
It remains to check the latter map induces an isomorphism of the source with the subsheaf $\om_{X/S}^{-2}(-\sum_{i\in I} D_i-2\sum_{j\in J} R_j)$.
This is a local question which is well known away from the punctures. Near the NS puncture this follows from Corollary \ref{NS-puncture-Tsc-isom-cor}, while near the Ramond puncture
this can be checked using standard coordinates (see Lemma \ref{local-descr-smooth-lem}) as in \cite[Prop.\ 3.12]{BHR}.
\end{proof}

In the case when the base is a point it is useful to rewrite the result in terms of the corresponding spin curve.

\begin{cor} Let $(X,P_\bullet,R_\bullet)$ be a smooth supercurve over $\C$ with the underlying spin curve $(C,L,p_\bullet,r_\bullet)$, where
$L^2\simeq \om_C(\sum r_j)$. Then
$$\AA_{X,P_\bullet,R_\bullet}^+\simeq \TT_C(-\sum p_i-\sum r_j), \ \ \AA_{X,P_\bullet,R_\bullet}^-\simeq \TT_C\ot L(-\sum p_i-\sum r_j).$$
\end{cor}

\subsection{Sheaf of infinitesimal automorphisms for stable supercurves}

The following local analysis of the sheaf  $\AA_{X,P_\bullet,R_\bullet}$ is from \cite{Deligne} (we corrected a misprint in the case of
the Ramond node).

\medskip

\noindent
{\it Ramond node}.

Locally near such a node, we have $\OO_X=\OO_C\oplus L$, where $L$ is locally free, $L^{\ot 2}\simeq \om_C$,
and
$$\AA_X=\TT_C\oplus L\ot \om_C^{-1}.$$

\medskip

\noindent
{\it NS node}.

Let $C=B_1\cup B_2$ be the two branches. Each $B_i$ is equipped with a square root $L_i$
of $\om_{B_i}$.
Then $\AA_X=\AA_1\oplus \AA_2$, with
$$\AA_i=\TT_{B_i}(-q_i)\oplus L_i\ot \TT_{B_i}(-q_i)$$

We can also determine the sheaf of infinitesimal automorphisms globally and
check the absence of infinitesimal automorphisms for stable supercurves by the standard count
(this is done in [Deligne] for the case of supercurves without punctures).

\begin{prop}\label{no-inf-aut-prop}
Let $(X,P_\bullet,R_\bullet)$ be a stable supercurve over $\C$ with the underlying stable spin curve $(C,L,p_\bullet,r_\bullet)$.
Let $\rho:\wt{C}\to C$ be the normalization with the induced spin structure $\wt{L}$
(see Sec.\ \ref{even-moduli-sec}), and let $(\wt{X},\wt{P}_\bullet,\wt{R}_\bullet)$ be the corresponding smooth supercurve with punctures.

\noindent
(i) One has natural a isomorphism
$$\AA^+_{X,P_\bullet,R_\bullet}\simeq \rho_*\AA^+_{\wt{X},\wt{P}_\bullet,\wt{R}_\bullet}\simeq \TT_C(-\sum p_i-\sum r_j)$$
and an exact sequence
$$0\to \AA^-_{X,P_\bullet,R_\bullet}\to \rho_*\AA^-_{\wt{X},\wt{P}_\bullet,\wt{R}_\bullet}\to \bigoplus_{q\text{ R-node}}\OO_q\to 0$$
where the summation is over all Ramond nodes.

\noindent
(ii) One has $H^0(X,\AA_{X,P_\bullet,R_\bullet})=0$.
\end{prop}

\begin{proof}

(i)  We have a natural morphism
$$\kappa:\AA_{X,P_\bullet,R_\bullet}\to \rho_*\AA_{\wt{X},\wt{P}_\bullet,\wt{R}_\bullet}$$
which is an isomorphism
away from the nodes. Thus, the assertion can be checked by a local computation near the nodes (so we can forget about the punctures).
Near an NS node we have
$$\AA^+_X\simeq \rho_*\TT_{\wt{C}}(-q_1-q_2), \ \ \AA^-_X\simeq \rho_*\wt{L}\ot\TT_{\wt{C}}(-q_1-q_2),$$
where $\{q_1,q_2\}\sub\wt{C}$ is the preimage of the node, so $\kappa$ is an isomorphism.

Near a Ramond node we have
$$\AA^+_X\simeq \TT_C\simeq \rho_*\TT_{\wt{C}}(-q_1-q_2), \ \ \AA^-_X\simeq L\ot \om_C^{-1},$$
whereas
$$\AA^-_{\wt{X}}\simeq \wt{L}\ot\om^{-1}_{\wt{C}}(-q_1-q_2)\simeq \rho^*(L\ot \om_C^{-1}).$$
Thus, $\kappa^+$ is an isomorphism, while $\kappa^-$ is an embedding with the cokernel of length 1 supported at the node.

\noindent
(ii)

Since global infinitesimal automorphisms of $X$ embed into those for $\wt{X}$,
it is enough to prove the assertion in the case when $C$ is smooth. We can also assume it is connected.

When $C$ is smooth we have
$$\AA_X^+=\TT_C(-\sum p_i-\sum r_j), \ \ \AA_X^-=L\ot \TT_C(-\sum p_i-\sum r_j).$$
We have $L^2\simeq \om_C(\sum r_j)$, so
$$\deg L=g-1+n_R/2,$$
$$\deg L\ot \TT_C(-\sum p_i-\sum r_j)=-(g-1)-n_{NS}-n_R/2, \ \
\deg \TT_C(-\sum p_i-\sum r_j)=-2(g-1)-n_{NS}-n_R.$$
Note that $\deg \TT_C(-\sum p_i-\sum r_j)<0$ by stability of the underlying pointed curve.
Hence,
$$\deg L\ot \TT_C(-\sum p_i-\sum r_j)=\frac{1}{2}\deg \TT_C(-\sum p_i-\sum r_j) - n_{NS}/2<0,$$
so $\AA_X^\pm$ do not have global sections.
\end{proof}

\section{Deformations}\label{def-sec}

In this section we will study the deformation functor of a stable supercurve (with punctures), proving that it is smooth.
For some background on deformation theory in the super context we refer to \cite{Vaintrob}.

\subsection{Smooth supercurves without Ramond punctures}\label{no-R-punct-def-sec}

Let $X_0$ be an affine (smooth) supercurve over $\C$.

\begin{lemma}\label{supeconf-field-pt}
For every $\C$-point $p\in X_0$, and every tangent vector $v_0\in T_pX_0$, there exists a superconformal
vector field $v$ on $X_0$ with $v(p)=v_0$.
\end{lemma}

\begin{proof} First, we claim that there is an exact sequence of sheaves with respect to \'etale topology
$$0\to \AA_{X_0,p}\to \AA_{X_0}\to i_*T_pX_0\to 0$$
where $\AA_{X_0}=\TT_{X_0}^{sc}$. Indeed, for this we need to check the surjectivity of the last arrow.
Locally in \'etale topology the distribution  $\DD\sub \TT_{X_0}$ has a generator $D=\pa_\th+\th\pa_z$ (see Lemma \ref{local-descr-smooth-lem}).
Then the vector fields $D-2\th D^2$ and $D^2$ are superconformal and restrict to a basis of $T_pX_0$.

Next, we observe that $\AA_{X_0,p}$ is isomorphic to a coherent sheaf on $X_0$ (see Proposition \ref{inf-aut-smooth-punct-prop}).
Hence, $H^1(X_0,\AA_{X_0,p})=0$ and we obtained the required surjectivity of the map $H^0(X_0,\AA_{X_0})\to T_pX_0$.
\end{proof}

We consider the category $\Art_{\C}$ of Artin local $\C$-(super)rings with the residue field $\C$.
Recall that a surjection $A\to B$ is called a {\it small extension} if $\fm\ker(A\to B)=0$,
where $\fm$ is the maximal ideal in $A$. Every surjection in $\Art_{\C}$ is a composition of a finite number of small extensions.

\begin{lemma}\label{smooth-curve-trivial-def-lem}
(i) Every deformation $X_A$ of $X_0$ over $A\in \Art_{\C}$ is isomorphic to a trivial deformation
$X_0\times \Spec(A)$, with the distribution induced by that on $X_0$.

\noindent
(ii) Given a surjection $A\to B$ in $\Art_{\C}$, and a deformation $X_A$ of $X_0$ over $A$, any superconformal automorphism
of $X_B=X_A\times_{\Spec A} \Spec B$ lifts to a superconformal automorphism of $X_A$.

\noindent
(iii) Analogs of (i) and (ii) hold for deformations of an affine supercurve $X_0$ with an NS-puncture $P_0\sub X_0$.

\noindent
(iv) For a surjection $A\to B$ in $\Art_{\C}$ any superconformal automorphism of $B(\!(z)\!)[\th]$ (with standard $\de$) lifts to a superconformal
automorphism of $A(\!(z)\!)[\th]$.
\end{lemma}

\begin{proof}
(i) Let $S_0$ (resp., $S_A$) be the algebra of functions on $X_0$ (resp., $X_A$). Let also $\fm\sub A$ be the
maximal ideal (which is nilpotent). We have an exact sequence
$$0\to \fm S_A\to S_A\to S_0\to 0.$$
In particular, $S_A^+$ is a nilpotent extension of $S_0^+$, hence, we can choose a section
$$\si^+:S_0^+\to S_A^+.$$
We know that $S_0^-$ is a locally free $S_0^+$-module of rank $1$. In particular, $S_0^+$ is projective,
so we can choose an $S_0^+$-module splitting
$$\si^-:S_0^-\to S_A^-$$
of the projection $S_A^-\to S_0^-$. Furthermore, we claim that $\si^-(S_0^-)\cdot \si^-(S_0^-)=0$ in $S_A$.
Indeed, this follows from the fact that locally $S_0^-$ is generated by one element.
Hence, $\si=(\si^+,\si^-):S_0\to S_A$ is a homomorphism of superalgebras.

From this we get a homomorphism $S_0\otimes_{\C} A\to S_A$ of $A$-algebras which induces isomorphism on
the graded quotients of the $\fm$-adic filtrations, hence, is an isomorphism.

Now let $\DD_0\sub\TT_{X_0/k}$ be the distribution giving the supercurve
structure on $X_0$, and let us set $X_A=X_0\times \Spec(A)$.
It remains to show that any distribution $\DD\sub \TT_{X_A/A}$
giving a supercurve structure over $A$, deforming $\DD_0$,
is isomorphic to the pull-back of $\DD_0\otimes A$ under some automorphism of $X_A$ trivial on $X_0$.

We can assume that for some ideal $I\sub A$, such that $\fm I=0$ (so $A\to A/I$ is a small extension),
the assertion holds for $A/I$,
so let
$$\DD\sub \TT_{X_A/A}\simeq \TT_{X_0/k}\otimes A$$
be a subbundle of rank $0|1$, which reduces to $\DD_0\otimes A/I$ over $A/I$.
Then $\DD$ corresponds to a homomorphism
$$f_\DD:\DD_0\to \TT_{X_0/k}/\DD_0\ot I.$$
On the other hand, an automorphism of $X_A$ trivial on $X_{A/I}$, corresponds to an $I$-valued vector field $v$
on $X_0$. It changes the distribution $\DD_0\otimes A$ to a distribution $\DD$ such that
the homomorphism $f_\DD$ is given by the Lie bracket with $v$. Hence, taking $v$ in $\DD_0\ot I$, we can
get any possible homomorphism $\DD_0\to \TT_{X_0/k}/\DD_0\ot I$.

\noindent
(ii) By (i) all deformations are trivial.
It is easy to check that the functor associating with $A$ the group of superconformal automorphisms of $X\times\Spec(A)$
is a deformation functor. Since we work over $\C$, by [Thm.\ 7.19]\cite{FM}, it is smooth.

\noindent
(iii) Let $\phi:S_A\to A$ be the homomorphism corresponding to the NS-puncture, deforming
$\phi_0:S_0\to k$. By (i), we can assume that $S_A=S_0\otimes_{\C} A$. We claim that there exists an automorphism of $S_A$
over $A$, trivial on $S_0$, and compatible with the superconformal structure,
transforming $\phi_0\ot A$ to $\phi$.

It is enough to check this assuming that
$\phi=\phi_0\ot A \mod I$, where $\fm I=0$. Then $\phi-\phi_0\ot A$ is given by a $\phi_0$-derivation $S_0\to I$,
i.e., by an $I$-valued tangent vector at $P_0\in X_0$. It remains to extend this tangent vector to an $I$-valued
superconformal vector field (see Lemma \ref{supeconf-field-pt})
and consider the corresponding automorphism of $X_A$.

The fact about automorphisms follows similarly to (ii) from smoothness of the corresponding group scheme.

\noindent
(iv) The proof is similar to (ii): the functor associating with $A$ the group of superconformal automorphisms of $A(\!(z)\!)[\th]$ is a deformation
functor. Since we work over $\C$, it is smooth.
\end{proof}

\subsection{Neighborhood of Ramond puncture}\label{R-punct-def-sec}

Let $X_0$ be an affine supercurve over $\C$ with one Ramond puncture $R_0\sub X_0$, and let
$\DD_0\sub \TT_{X_0/k}$ be the structure distribution (so that \eqref{Ramond-D2-isom} is an isomorphism).

\begin{lemma}\label{Ramond-surj-lem}
The natural morphism of sheaves
$$\TT_{X_0/k}\to\und{\Hom}_{\OO}(\DD_0,\TT_{X_0/k}/\DD_0): v\mapsto (v_0\mapsto [v,v_0]\mod\DD_0)$$
is surjective with respec to \'etale topology. Hence, we have an exact sequence of sheaves in classical topology
$$0\to \TT^{sc}_{X_0/k}\to \TT_{X_0/k}\to \und{\Hom}_{\OO}(\DD_0,\TT_{X_0/k}/\DD_0)\to 0$$
and the induced map on global sections
$$H^0(X_0,\TT_{X_0/k})\to \Hom_{\OO}(\DD_0,\TT_{X_0/k}/\DD_0)$$
is surjective
\end{lemma}

\begin{proof} Locally in \'etale topology we can assume that $\DD_0$ be generated by $v_0=\pa_\th+\th z\pa_z$ (see Lemma \ref{local-descr-smooth-lem}).
Then $\pa_z$ projects to a basis of $\TT_{X_0/k}$.
Thus, it is enough to check that for every function $a=a(z,\th)$ there exists $v$ with
$$[v,v_0]=a\pa_z\mod\DD_0.$$
We can represent every $a$ in the form $a=c_0+c_1\th+bz$ for some function $b$ and some constants
$c_0,c_1$. It remains to note that
$$[(c_0\th+c_1)\pa_z+bv_0,v_0]=(\pm c_0+c_1\th+2bz)\pa_z.$$

The last statement is a consequence of the vanishing $H^1(X_0,\TT^{sc}_{X_0/k})=0$ which itself follows from the fact that
$\TT^{sc}_{X_0}$ is isomorphic to a coherent sheaf (see Proposition \ref{inf-aut-smooth-punct-prop}).
\end{proof}

We have the following analog of Lemma \ref{smooth-curve-trivial-def-lem}.

\begin{lemma} (i) Every deformation $(X_A,R_A)$ of $(X_0,R_0)$ over $A\in \Art_{\C}$ is isomorphic to the trivial deformation
$(X_0\times \Spec(A), R_0\times\Spec(A))$, with the distribution induced by that on $X_0$.

\noindent
(ii) Given a surjection $A\to B$ in $\Art_{\C}$, and a deformation $(X_A,R_A)$ of $(X_0,R_0)$ over $A$, any automorphism
of $(X_B,X_R)$ (obtained by the base change to $B$) lifts to an automorphism of $(X_A,R_A)$.
\end{lemma}

\begin{proof}
(i)

As we have seen in Lemma \ref{smooth-curve-trivial-def-lem}(i),
we can assume that $X_A=X_0\times \Spec(A)$. We have a distribution $\DD_A\sub \TT_{X_A/A}$ deforming $\DD_0$.

It is enough to check that $\DD_A$ is obtained from $\DD_0\ot A$
by an automorphism of $X_A$, trivial on $X_0$.
We can assume that for some ideal $I\sub A$, such that $\fm I=0$, the assertion holds for $A/I$,
so $\DD_A$ corresponds to a homomorphism
$$f_\DD:\DD_0\to \TT_{X_0/k}/\DD_0\ot I.$$
Now the assertion follows from Lemma \ref{Ramond-surj-lem}.

\noindent
(ii) As before, this follows from (i) and from smoothness of the corresponding group scheme.
\end{proof}

\subsection{Unobstructedness}

Recall that a morphism of deformation functors $\Def_1\to \Def_2$ is called {\it smooth} if for every small extension $B\to A$ in $\Art_{\C}$,
the natural map
$$\Def_1(B)\to \Def_1(A)\times_{\Def_2(A)}\Def_2(B)$$
is surjective.

\begin{lemma}\label{aff-formal-node-smooth-lem}
(i) Let $X_0$ be an affine stable supercurve over $\C$ with a single NS node $q$ and let $\Def(\hat{\OO})$ denote the functor of deformations of
the completion of the local ring of $X_0$ at $q$,
$$\hat{\OO}\simeq \C[\![z_1,z_2]\!][\th_1,\th_2]/(z_1z_2,\th_1z_2,\th_2z_1,\th_1\th_2).$$
Assume that there exists a function $f_0$ on $X_0$ such that in the formal neighborhood of $q$ one has
$f_0\equiv z_1+z_2\mod (z_1,z_2,\th_1,\th_2)^2$, and the principal open affine $D(f_0)\sub X_0$ coincides with $X_0\setminus\{q\}$.
Then the natural morphism of deformation functors
$$\Def(X_0)\to \Def(\hat{\OO})$$
is smooth.

\noindent
(ii) Similar assertions hold for am affine stable supercurve with a single Ramond node.
\end{lemma}

\begin{proof}
(i) First, let us introduce the following notation. For $A\in \Art_{\C}$ and $t\in \fm_A$, we set
$$\hat{\OO}_{A,t}:=A[\![z_1,z_2]\!][\th_1,\th_2]/(z_1z_2+t^2,z_1\th_2-t\th_1,z_2\th_1+t\th_2,\th_1\th_2).$$

\noindent
{\bf Step 1}. Let $A\in \Art_{\C}$, $t$ be an element such that $t^n=0$. Let us consider a homomorphism of $A$-algebras,
$$\ga:\hat{\OO}_{A,t}\to A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]: \phi(z_1,z_2,\th_1,\th_2)\mapsto (\phi(z_1,-t^2/z_2,\th_1,t\th_1/z_1),
\phi(-t^2/z_2,-t\th_2/z_2,z_2,\th_2)).$$
Note that this is well-defined since $t^n=0$. It is also easy to see that $\ga$ is injective,
and the elements
$$(z_1^{-i},z_1^{-i}\th_1,z_2^{-i},z_2^{-i}\th_2)_{i\le 0}$$
project to a basis of the quotient by the image of $\ga$, as an $A$-module.
We claim that for any element $f\in \hat{\OO}_{A,t}$ such that $f\equiv z_1+z_2\mod (z_1,z_2,\th_1,\th_2)^2$,
$\ga$ is identified with the localization map
$$\hat{\OO}_{A,t}\to \hat{\OO}_{A,t}[f^{-1}].$$

Indeed, since the multiplication by $f$ is invertible on $A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]$, it is enough to check that
for every $(p,q)\in A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]$ one has $f^N(p,q)\in \im(\ga)$ for sufficiently large $N$.
But this follows from the inclusion
$$z_1^nA[\![z_1]\!][\th_1]\oplus z_2^nA[\![z_2]\!][\th_2]\sub \im(\ga).$$

Note that this entire picture is obtained by the change of coefficients $\C[T]/(T^n)\to A$ from the standard picture described in Sec.\ \ref{nodes-def-sec}.
It follows that we have a structure derivation
$$\de:\hat{\OO}_{A,t}\to \om_{\hat{\OO}_{A,t}/A},$$
such that after inverting $z_1$ or $z_2$ it is given by the formula \eqref{NS-miniversal-de-formula}.

\noindent
{\bf Step 2}. Let $X_B$ be a deformation of $X_0$ over $B\in \Art_{\C}$. Then for any function $f\in \OO(X_B)$, lifting $f_0$,
the principal open affine $D(f)\sub X_B$ is the trivial deformation of $D(f_0)=X_0\setminus\{q\}$ over $B$.
Furthermore, there exist an isomorphism of supercurves $D(f)\simeq \Spec(B)\times D(f_0)$, and a superconformal isomorphism
\begin{equation}\label{completion-XB-t-isom}
\hat{\OO}_{X_B,q}\simeq \hat{\OO}_{B,t},
\end{equation}
for some element $t$ of the maximal ideal in $B$, so that we have a cartesian diagram
\begin{diagram}
\OO(X_B)&\rTo{}& B\ot \OO(D(f_0))\\
\dTo{}&&\dTo{r_B}\\
\hat{\OO}_{B,t}&\rTo{\ga}&B(\!(z_1)\!)[\th_1]\oplus B(\!(z_2)\!)[\th_2]
\end{diagram}
Here $r_B$ comes from a certain isomorphism $B\ot \widehat{\OO(D(f_0))}\simeq B(\!(z_1)\!)[\th_1]\oplus B(\!(z_2)\!)[\th_2]$,
where we equip $\OO(D(f_0))$ with the topology using powers of $I_q$, the ideal of a node in $\OO(X_B)$, and consider the completion.

Indeed, it is clear that $D(f)$ is a deformation of $D(f_0)$ as a supercurve.
Since $D(f_0)$ is a smooth affine supercurve, by Lemma \ref{smooth-curve-trivial-def-lem}(i) all its deformations are trivial.
The existence of a superconformal isomorphism \eqref{completion-XB-t-isom} follows from Theorem \ref{miniversal-Deligne-thm}.
Finally, by Step 1, we have
$$B(\!(z_1)\!)[\th_1]\oplus B(\!(z_2)\!)[\th_2]\simeq \hat{\OO}_{B,t}[f^{-1}]\simeq \hat{\OO}_{B,t}\ot_{\OO(X_B)}\OO(X_B)[f^{-1}]\simeq
\hat{\OO}_{B,t}\ot_{\OO(X_B)} (B\ot \OO(D(f_0))),$$
which is isomorphic to the completion of $B\ot \OO(D(f_0))$ with respect to the $I_q$-adic topology.

\noindent
{\bf Step 3}. Now let $A\to B$ be a small extension in $\Art_{\C}$.

We claim that there exists a homomorphism
$$r_A:A\ot \OO(D(f_0))\to A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]$$
lifting $r_B$ and compatible with derivations. Now for an element
$\wt{t}\in A$ lifting $t\in B$,
let us define the ring $\wt{\OO}$ as the fibered product
\begin{equation}\label{def-A-fibered-product-diagram}
\begin{diagram}
\wt{\OO}&\rTo{}& A\ot \OO(D(f_0))\\
\dTo{}&&\dTo{r_A}\\
\hat{\OO}_{A,\wt{t}}&\rTo{\ga}&A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]
\end{diagram}
\end{equation}
Then we claim that $\wt{\OO}=\OO(X_A)$ for some deformation $X_A$ of $X_0$ inducing $X_B$ (so there is also a derivation $\de$ on $X_A$).

First, we have a superconformal isomorphism
$$\a_0:\widehat{\OO(D(f_0))}\simeq \C(\!(z_1)\!)[\th_1]\oplus \C(\!(z_2)\!)[\th_2].$$
Hence, the map $r_B$ corresponds to the composition of  superconformal isomorphisms
$$B\ot \widehat{\OO(D(f_0))}\rTo{B\ot \a_0} B(\!(z_1)\!)[\th_1]\oplus B(\!(z_2)\!)[\th_2]\rTo{\a_B} B(\!(z_1)\!)[\th_1]\oplus B(\!(z_2)\!)[\th_2]$$
for some $\a_B$. Now we define $r_A$ using the composition of superconformal isomorphisms
$$A\ot \widehat{\OO(D(f_0))}\rTo{A\ot \a_0} A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2]\rTo{\a_A} A(\!(z_1)\!)[\th_1]\oplus A(\!(z_2)\!)[\th_2],$$
where $\a_A$ is some lift of $\a_B$. We know that such a lift exists by Lemma \ref{smooth-curve-trivial-def-lem}(iv).

Since the lower horizontal arrow in \eqref{def-A-fibered-product-diagram} is a split embedding of $A$-modules, so is the upper horizontal arrow.
Hence, $\wt{\OO}$ is a projective $A$-module. Applying the reduction $?\ot_A B$ to our diagram we recover the
cartesian diagram from Step 2, so $\wt{\OO}\ot_A B\simeq \OO(X_B)$.

Finally, we observe that we have a cartesian diagram of the dualizing sheaves parallel to diagram \eqref{def-A-fibered-product-diagram}.
Since both $r_A$ and $\ga$ are compatible with the derivations, we get a
a well defined derivation $\de:\wt{\OO}\to \om_{\wt{\OO}/A}$.

\noindent
(ii) The proof is similar to (i).
\end{proof}

\begin{prop}\label{global-def-prop}
Let $(X,P_\bullet,R_\bullet)$ be a stable supercurve with punctures over $\C$. Let $q_1,\ldots,q_m$ be all the nodes of $X$.

Then the morphism of deformation functors
$$\Def(X,P_\bullet,R_\bullet)\to \prod_{i=1}^m \Def(\OO_{X,q_i})$$ is smooth.
The induced morphism of tangent spaces fits into an exact sequence
\begin{equation}\label{def-curve-nodes-ex-seq}
0\to H^1(X,\AA_{X,P_\bullet,R_\bullet})\to T_{\Def(X,P_\bullet,R_\bullet)}\to \bigoplus_{i=1}^m T_{\Def(\hat{\OO}_{X,q_i})}\to 0.
\end{equation}
\end{prop}

\begin{proof}
The main idea is that away from the nodes, locally all deformations are trivial, so everything boils down to some
cohomology and we use the vanishing of $H^2$.

Let us cover $X$ by affine opens $U_i$ such that for $i=1,\ldots,m$, $U_i$ contains $q_i$ and does not contain other
nodes or punctures, while for $i>m$, $U_i$'s lies in the smooth locus and
contains at most one R-puncture or NS-puncture.

Now suppose $B\to B/I=A$ is a small extension in $\Art_{\C}$,
and we are given a deformation $X_A$ of $X$ over $A$ (with all the punctures), and
liftings of the induced deformations of $\hat{\OO}_{X,q_i}$ to deformations of singularities over $B$.
By Lemma \ref{aff-formal-node-smooth-lem}, for $i=1,\ldots,m$, we have a deformation
$U_{iB}$ of $U_i$ over $B$, inducing the given deformation of the singularity and
such that the induced deformation $U_{iA}$ over $A$ comes from $X_A$.
By the results of Sec.\ \ref{no-R-punct-def-sec} and \ref{R-punct-def-sec} all the local deformations
$U_{iA}$, for $i>m$, are trivial, so we can
lift them to trivial deformations $U_{iB}$ over $B$. Furthermore, all the intersections $U_{ij}$ are smooth and contain
no punctures, so all their deformations are trivial, and we have some isomorphisms $\a_{ijB}$ between
the deformations of $U_{ij}$ over $B$, induced from $U_{iB}$ and $U_{jB}$.
The induced isomorphisms $\a_{ijA}$ may differ from the ones coming from $X_A$, but we can correct
each $\a_{ijB}$ using Lemma \ref{smooth-curve-trivial-def-lem}(ii) to make sure that $\a_{ijA}$ are the ones coming from $X_A$.

Thus, considering $\a_{kiB}\a_{jkB}\a_{ijB}$ over triple intersections we get a $2$-cocycle with values in
$I\ot \AA_{X,P_\bullet,R_\bullet}$. Since the corresponding $H^2$ vanishes, this $2$-cocycle is a coboundary.
Hence, we can correct $\a_{ijB}$, so that they become compatible on triple intersections (without changing $\a_{ijA}$).
Thus, our data will give a deformation $X_B$ (with all the punctures).

By smoothness, we have surjectivity in sequence \eqref{def-curve-nodes-ex-seq}. It remains to observe that $H^1(X,\AA_{X,P_\bullet,R_\bullet})$
is identified with the space of locally trivial infinitesimal deformations.
\end{proof}

Using the fact that the deformations of nodal singularities of supercurves are smooth (see Sec. \ref{nodes-def-sec}),
we derive the following

\begin{cor}\label{smooth-def-functor-cor}
The functor $\Def(X,P_\bullet,R_\bullet)$ is smooth.
\end{cor}

\section{Square of the relative canonical bundle}\label{square-sec}

In this section we will define the line bundle $\om^2_{X/S}$ on certain families of stable supercurves, as an extension from the smooth locus.
We show that it is useful in finding an ample line bundle over $X$, as well as due to its relation to the sheaf on infinitesimal automorphisms.

\subsection{Local freeness and ampleness}

It is well known that for a stable curve with punctures $(C,p_1,\ldots,p_n)$ over $S_0$, the line bundle
$\om_{C/S_0}^{log}=\om_{C/S_0}(p_1+\ldots+p_n)$ is relatively ample.
Our goal is to find an analogous construction for families of stable supercurves.
The problem is that for a stable supercurve $X/S$ the relative dualizing sheaf $\om_{X/S}$ is not necessarily locally free.

We show that this problem can be solved by extending $\om^2_{X/S}$ from the smooth locus.

\begin{theorem}\label{can-square-thm}
Let $f:X\to S$ be a family of prestable supercurves inducing a smooth morphism to the base of the miniversal deformation of every node
in a fiber. Let $j:U\hra X$ be the open complement to the locus of nodes in fibers. Then for any $n\in \Z$, the sheaf
$\om_{X/S}^{2n}:=j_*\om_{U/S}^{\ot 2n}$ is a line bundle on $X$. For a morphism $u:S'\to S$ such that the induced family $X'\to S'$
also satisfies the above assumption, there is a natural isomorphism
\begin{equation}\label{om-2n-base-change-eq}
v^*\om_{X/S}^{2n}\rTo{\sim} \om_{X'/S'}^{2n},
\end{equation}
where $v:X'\to X$ is the induced morphism. In the case when $S$ is even, and $(C,L)$ is the underlying family of curves with spin structures, we have
$$\om_{X/S}^{2n}|_C\simeq \om_{C/S}^n\oplus \om_{C/S}^n\ot L=p^*\om_{C/S}^n,$$
where $p:X\to C$ is the natural projection.
In particular,
$$\om_{X/S}^{2n}|_C\simeq \om_{C/S}^n.$$
\end{theorem}

\begin{proof}
We claim that under our assumptions $X$ is Cohen-Macaulay near the nodes. Indeed, by assumption, $S$ is smooth near
every point $s$ with singular fiber $X_s$. Since the morphism $f$ is Cohen-Macaulay, the claim follows from
\cite[Lem.\ 37.20.4]{Stacks}.
Since the complement to $U$ has codimension $\ge 2$ in $X$, by the cohomological interpretation of depth,
we see that the natural map
$$\OO_X\to j_*\OO_U$$
is an isomorphism. We will use this fact below.

Let us first prove that $j_*\om_{U/S}^{\ot 2}$ is a line bundle
for miniversal deformations of the nodes, i.e., when $S=\Spec k[t]$ and $X$ is given as in
\ref{univ-def-node-I} and \eqref{univ-def-node-II}.

For a Ramond node, $\om_{X/S}$ is a line bundle, hence, $\om_{X/S}^{\ot 2}$ is also a line bundle, and
$$\om_{X/S}^{\ot 2}\simeq j_*\om_{U/S}^{\ot 2}.$$

For an NS node, the total space $C$ of the underlying usual family of curves is the quadratic cone
$xy=-t^2$, and $U^{\red}$ is the complement to the singular point $q$ in $C$. Furthermore, we have
$$\OO_X=\OO_C\oplus L, \ \ \om_{X/S}\simeq L\oplus \om_{C/S},$$
where $L$ is a CM-sheaf of rank $1$ over $C$. Over $U$ we have
$$\om_{U/S}^{\ot 2}\simeq \om_{U^{\red}/S}\oplus \om_{U^{\red}/S}\ot L|_U.$$
We can compute the push-forward $j_*$ componentwise. Since the complement of $U$ has codimension $\ge 2$,
$\om_{C/S}$ is locally free and $L$ is CM, we get
$$j_*\om_{U/S}^{\ot 2}\simeq \om_{C/S}\oplus \om_{C/S}\ot L,$$
which is just the pull-back of $\om_{C/S}$ under the natural projection $X\to C$.
Hence, $j_*\om_{U/S}^{\ot 2}$ is locally free of rank $1$.

To prove the result in the general case, i.e., for an arbitary family $X/S$ inducing a smooth morphism to the base of the miniversal deformation of every node,
we observe that the question is local, and an \'etale neighborhood $B$ of $X$ will have a smooth morphism $t:B\to X_0$
to the miniversal deformation of the node $X_0/S_0$, so that $B\cap U=t^{-1}(U_0)$, where $U_0\sub X_0$ is the complement to the node. Then the base change morphism
$$t^*j_{0*}\om_{U_0/S_0}^2\to j_*\om_{U/S}^2$$
is an isomorphism, hence, $j_*\om_{U/S}^2$ is locally free.
The isomorphism \eqref{om-2n-base-change-eq} follows from this and from the compatilibity of the base change morphisms for the maps to $X_0$
from \'etale neighborhoods of nodes in $X'$ and in $X$.
\end{proof}

Now we recall that for every NS-puncture $P_i$,
there is a canonical divisor $D_i$ with the same support (see Sec.\ \ref{NS-correspondence-sec}). So, we can define the line bundle on $X$,
$$\LL_{X/S}:=\om_{X/S}^{\ot 2}(\sum D_i+\sum R_j).$$

Let $f:X\to S$ be a proper morphism of Noetherian superschemes.

\begin{definition}
We say that a line bundle (of rank $1|0$) $L$ over $X$ is {\it strongly relatively ample over } $S$

if there exists $n>0$,

a supervector bundle $\EE$ over $S$ and a closed embedding $\phi:X\to \P(\EE)$ over $S$,
such that $L^n\simeq\phi^*\OO(1)$.

\end{definition}

For a superscheme $S$ we denote by $S_{\bos}$ the usual scheme with the same underlying topological space as $S$ and with the sheaf of rings
$\OO_S/\NN_S$, where $\NN_S$ is the ideal generated by odd functions.

\begin{theorem}\label{ample-thm}
Let $f:X\to S$ be a family of stable supercurves inducing a smooth morphism to the base of the miniversal deformation of every node
in a fiber. Then the line bundle $\LL_{X/S}$ is strongly relatively ample.
\end{theorem}

\begin{proof} Let $C\to S_0$ be the underlying usual family of stable curves over $S_0=S_{\bos}$.
Since $f$ is flat, by Proposition \ref{ample-crit-prop} of the Appendix, it suffices to prove that $L:=\LL_{X/S}|_C$ is strongly relatively ample over $S_0$.
Since $L$ is a line bundle, the natural map
$$L_{X/S}\to j_{\bos *}j_{\bos}^*L_{X/S}$$
is an isomorphism, where $j_{\bos}:U_{\bos}\to C$ is the embedding of the complement to the nodes.
We have
$$j_{\bos}^*L_{X/S}\simeq \om_{U_{\bos}/S_0}(\sum D_{i \bos}+\sum R_{j \bos}).$$
Hence,
$$L_{X/S}\simeq \om_{C/S_0}(\sum D_{i \bos}+\sum R_{j \bos}),$$
which is relatively ample by a classical result on stable curves (see \cite[Thm.\ 1.2]{DM}, \cite[Lem.\ 6.1]{ACG}).
\end{proof}

\subsection{The sheaf of infinitesimal symmetries}

Now we can extend Proposition \ref{inf-aut-smooth-punct-prop} to the case of stable supercurves.

\begin{theorem}\label{inf-aut-thm}
Let $(X,P_\bullet,R_\bullet)\to S$ be a family stable supercurve with punctures, where $(P_i)_{i\in I}$ are NS punctures
and $(R_j)_{j\in J}$ are Ramond punctures, inducing a smooth morphism to the base of the miniversal deformation of every node in a fiber.
Then one has a natural isomorphism
$$\AA_{(X,P_\bullet,R_\bullet)/S}\simeq \om_{X/S}^{-2}(-\sum_{i\in I} D_i-2\sum_{j\in J} R_j),$$
where $\om_{X/S}^{-2}$ is the line bundle defined in Theorem \ref{can-square-thm}
and $D_i\sub X$ is the divisor associated with the NS puncture $P_i$ (see Sec.\ \ref{NS-correspondence-sec}).
\end{theorem}

\begin{proof}
Over the smooth locus this holds by Proposition \ref{inf-aut-smooth-punct-prop}.

Now let us show that the natural map
$$\AA_{X/S}\to j_*\AA_{U/S},$$

where $j:U\to X$ is the embedding of the smooth locus, is an isomorphism.
Since the condition on $v\in \TT_{X/S}$ to belong to

$\AA_{X/S}$ can be imposed only over the smooth locus,
it is enough to check that the natural map
$$\TT_{X/S}\to j_*\TT_{U/S}$$
is an isomorphism. But this follows immediately from the fact that $j_*\TT_{U/S}$ can be identified with relative
derivations of $j_*\OO_U$ and the isomorphism $j_*\OO_U\simeq \OO_X$.
\end{proof}

\begin{remark}
Note that neither $\TT_{X/S}$ nor $\AA_{X/S}$ are compatible with the base change. So in the situation of Theorem \ref{inf-aut-thm} the restriction of
$\AA_{X/S}$ to a fiber $X_s$, which is a stable supercurve with at least one node, is a line bundle on $X_s$, whereas $\AA_{X_s}$ is not
(since $T_C$ is not locally free for a nodal curve $C$).
\end{remark}

\section{Proof of Theorem A}\label{proof-thm}

To prove Theorem $A$ we need to check that $\ov{\SS}_{g,n_{NS},n_R}$ is a stack with representable diagonal, with an \'etale atlas,
and that it is smooth and proper over $\C$.

\subsection{$\ov{\SS}_{g,n_{NS},n_R}$ is a limit preserving stack}

A family of stable supercurves is given by a superscheme $X\to S$, smooth of dimension $1|1$,
together with a morphism $\de:\Om_{X/S}\to \om_{X/S}$ and sections $p_i:S\to X$ (the R-punctures can
be recovered from $\de$). The fact that isomorphisms between two families over $S$ form a sheaf in \'etale topology
 is proved in the standard way (and also follows from the representability proved below).
 The \'etale descent for such families follows as in the classical case from the existence of the relatively ample bundle which we proved
in Theorem \ref{ample-thm} (see  \cite[Sec.\ 4.3.3]{FGA-exp}).

It is also clear that our stack $\MM$ is limit preserving: if $S=\Spec(A)$ (where $A$ is Noetherian),
with $A=\varinjlim_i A_i$, then $\MM(A)\simeq \varinjlim_i \MM(A_i)$.

\subsection{Representability of the diagonal}

We need to check that for a pair of families of stable supercurves
$X\to S$ and $Y\to T$, there is an algebraic space locally of finite type over $\C$, classifying isomorphisms
between $X_s$ and $Y_t$ as abstract superschemes. Passing to the induced families over $S\times T$, we can assume that
we have two families $X\to S$ and $Y\to S$ over the same base, and we need to check
that the functor $\Isom(X,Y)$ is representable by an algebraic space over $S$. In fact, we will check that it is representable by a superscheme over $S$.

For this we can use relative projectivity of the morphisms $X\to S$ and $Y\to S$ which holds by Theorem \ref{ample-thm}
and the standard approach via the Hilbert (super)schemes (the construction of Hilbert superschemes for
projective morphisms of superschemes is discussed in \cite[Sec.\ 4]{Hilbert-schemes} and in \cite[Sec.\ 7]{MZ}).
Namely, as in the classical case, the idea is that to an isomorphism $X_s\to Y_s$
we can associate its graph, which is a closed subscheme in $X_s\times Y_s$.
In more detail, first, let $\HH$ be the relative Hilbert superscheme over $S$ parametrizing subschemes in $X_s\times Y_s$
(with the same Hilbert series as the Hilbert series of $X_s$ with respect to a large power of $\LL_{X/S}|_{X_s}$).
Let
$$Z\sub X\times_S\times Y\times_S \HH$$
be the universal subscheme.
Let us consider the projections
$$p_X:Z\to X\times_S \HH, \ \ p_Y:Z\to Y\times_S \HH.$$
Then there is a universal open subscheme $\HH_1\sub \HH$
over which $p_X$ and $p_Y$ become isomorphisms (this is proved as in \cite[Thm.\ 5.22]{FGA-exp}).
Then it is easy to see that $\HH_1$ represents $\Isom_{\sSch/S}(X,Y)$.

Next, assume that our stable supercurves $X$ and $Y$ over the same base are equipped with the matching number of punctures of each type,
$(P_i^X, R_j^X)$ and $(P_i^Y, R_j^Y)$.
Let $Z\sub X\times_S\times Y\times_S\HH_1$ be the graph of the universal isomorphism.
Then we have the induced sections $Z\times_X P_i^X$ and $Z\times_Y P_i^Y$ of the family $Z\to \HH_1$,
and induced divisors $Z\times_X R_j^X$ and $Z\times_Y R_j^Y$ in $Z$, which are smooth of dimension $0|1$ over $\HH_1$. Let us set $P_X$ (resp.,$P_Y$) be the subscheme of $Z$ obtained as the disjoint union of these subschemes pulled back from $X$ (resp., $Y$). Then there exists the largest closed subscheme
$\HH_2\sub\HH_1$ such that $P_X$ and $P_Y$ coincide over $\HH_2$.
This is a consequence of the following easy Lemma.

\begin{lemma}\label{vanishing-closed-subscheme-lem}
(a) Let $f:F_1\to F_2$ be a morphism of coherent sheaves over a superscheme $Z$, and let $Z\to S$ be a morphism.
Assume that $F_1$ and $F_2$ are flat over $S$. Then there exists the largest closed subscheme
$T\sub S$ such that $f=0$ over $T$.

\noindent
(b) Let $R$ and $R'$ be subschemes in a superscheme $Z$, and let $Z\to S$ be a flat morphism such that both $R$ and $R'$ are flat over $S$.
Then there exists the largest closed subscheme
$T\sub S$ such that $R=R'$ over $T$.
\end{lemma}

\begin{proof}
(a) Let $Z_f\sub Z$ be the closed subscheme corresponding to the ideal $\ker(f:\OO_Z\to \und{\Hom}(F_1,F_2))$. Then $T$ is the schematic image of $Z_f$.

\noindent
(b) We apply (a) to the morphisms $I_R\to \OO_Z/I_{R'}$ and $I_{R'}\to \OO_Z/I_R$ and observe that these morphisms vanish if and only if $R=R'$.
\end{proof}

Finally, we can define a closed subscheme $\HH_{sc}\sub \HH$ that corresponds to superconformal isomorphisms.
To this end we observe that we can pull back to $Z$ both morphisms $\de_X:\Om_{X/S}\to \om_{X/S}(R_X)$ and $\de_Y:\Om_{Y/S}\to \om_{Y/S}(R_Y)$,
and then apply Lemma \ref{vanishing-closed-subscheme-lem} to the difference.
Then $\HH_{sc}$ represents the sheaf of isomorphisms between families $X$ and $Y$ as stable supercurves with punctures.

\subsection{Construction of an \'etale atlas}

The main point is to use the existence of the (purely even) Deligne-Mumford stack $\ov{\SS}=\ov{\SS}_{g,n_{NS},n_R}$ parametrizing
stable curves with spin-structures (and punctures), constructed in \cite{AJ}.

For every stable supercurve $X_0$ over $\C$, we can find a family $\pi:\XX_0\to B_0$ of stable supercurves (with punctures) over an affine even base $B_0$,
and a $\C$-point $b\in B_0$ such that we get $X_0$ as a fiber of $\XX_0$ over $b$,
such that the corresponding map $B_0\to \ov{\SS}$ is \'etale.

Let us consider the corresponding bundle over $B_0$,
$$\EE:=R^1\pi_*(\AA^-)=R^1\pi_*(\om_{\CC/B_0}^{-1}\ot \LL(-\sum p_i-\sum r_i)),$$
where $\CC\sub \XX_0$ is the corresponding usual curve over $B_0$ with the relative spin-structure $\LL$
(recall that $\AA^-$ denotes the sheaf of odd infinitesimal automorphisms of $\XX_0/B_0$). Note that the fact that $\EE$ is
locally free follows from the vanishing of $\pi_*(\AA^-)$, i.e., from the absence of odd infinitesimal automorphisms of $\XX_0/B_0$.

We define the superbase $B$ as
$$B=\Spec_{B_0}({\bigwedge}^\bullet \Pi\EE^\vee).$$
In other words, we view $\EE^\vee$ as extra odd coordinates.
Our goal is to extend $\XX_0$ to a family $\XX\to B$ (possibly after changing $B_0$ to an \'etale neighborhood of $b$).

Let $q_1,\ldots,q_N$ be a finite number of points in $C_0$, including all nodes, $q_1,\ldots,q_n$,
such that there is an ample effective divisor $D$ supported on $\{q_1,\ldots,q_N\}$, and $q_i$ are distinct from all the punctures.
Let also $q'_1,\ldots,q'_m$ be a set of smooth points in $C_0$, distinct from all the punctures and from $q_i$'s, such that
$D'=q'_1+\ldots+q'_m$ is ample.

We claim that replacing $B_0$ by an \'etale neighborhood of $b$, we can assume
the existence of relatively ample effective Cartier divisors $\DD$ and $\DD'$ in $\CC_0$, extending $D$ and $D'$,
such that $\DD\cap \DD'=\emptyset$.

Therefore, we have an affine cover of $\XX_0$ by two open sets, $\UU_0$ defined as the complement to $D'$ and $\UU_1$ defined as the complement to $D$.
Now, let us consider the tautological cohomology class
$$[c]\in H^0(B_0,\EE^\vee\ot R^1\pi_*(\AA^-))\simeq H^1(\XX_0,\pi^*\EE^\vee\ot \AA^-).
$$
We can represent it by a Cech cocycle
$$c\in H^0(\UU_0\cap \UU_1,\pi^*\EE^\vee\ot \AA^-).$$

Now we observe that there is a natural morphism of sheaves
$$\om_{\CC/B_0}^{-1}\ot \LL(-\sum p_i-\sum r_i)=\AA^-\to \TT_{\XX_0/B_0,P_\bullet,R_\bullet}^-.$$
Hence, we can view odd sections of $\pi^*\EE^\vee\ot \AA^-$ as an even derivation
$$\OO_{\XX_0}\to \Pi\pi^*\EE^\vee$$
(relative to $B_0$). Thus, $c$ gives rise to such a derivation over $\UU_0\cap \UU_1$.
Hence, we have a homomorphism
$$\exp(c):\OO\to {\bigwedge}^\bullet_{\OO} \Pi\pi^*\EE^\vee,$$
over $\UU_0\cap \UU_1$, reducing to identity modulo ${\bigwedge}^{\ge 1}$.
We can extend $\exp(c)$ to the automorphism $\wt{c}$ of
${\bigwedge}^\bullet_{\OO_{\XX_0}} \Pi\pi^*\EE^\vee$, identical on $\pi^{-1}{\bigwedge}^\bullet_{\OO_{B_0}}\Pi\EE^\vee$.

Now we define $\XX$ by gluing
$$\UU_i\times_{B_0} B\simeq \Spec({\bigwedge}^\bullet_{\OO} \Pi\pi^*\EE^\vee|_{\UU_i}), \ \ i=0,1,$$
using $\wt{c}$ as an automorphism of $(\UU_0\cap\UU_1)\times_{B_0} B$.

Note that since on a smooth locus $c$ acts by a derivation preserving the structure distribution $\DD\sub \TT$,
the same is true for $\wt{c}$. Hence, $\XX$ has a natural structure of a stable supercurve over $B$ (see Lemma \ref{isom-stable-lem}).

We claim that the corresponding map from $B$ to the moduli stack $\SS\MM$ of supercurves is \'etale near $b$.
Indeed, it is enough to check that the induced map on tangent spaces at $b$ is an isomorphism.
The tangent space $T_bB$ to $B$ at $b$ is given by
$$(T_bB)^+=T_bB_0, \ \ (T_bB)^-=\EE_b=H^1(X_0,\AA^-_{X_0}).$$
The fact that the map on even tangent spaces is an isomorphism follows from the assumption that $B_0\to \SS$ is \'etale.
The map on odd tangent spaces corresponds to the natural map
$$H^1(X_0,\AA^-_{X_0})\to (T_b\SS\MM)^-.$$
The fact that it is an isomorphism follows from the exact sequence \eqref{def-curve-nodes-ex-seq} since the last term of this sequence is purely even.

\subsection{Properties of the stack of stable supercurves}

Corollary \ref{smooth-def-functor-cor} shows that the stack of stable supercurves is smooth. Since it is of finite type (for each fixed genus and fixed number of punctures),
the fact that it is proper can be checked for its even part, i.e., for the stack of stable curves with spin structures. But this is known due to works of Cornalba \cite{Cornalba}
and Jarvis \cite{Jarvis}.

\section{Kodaira-Spencer map}\label{KS-map-sec}

In this section we will study the behavior of the Kodaira-Spencer map in degenerating families of supercurves where the limiting curve acquires one NS or Ramond
node. This will later help us to calculate the canonical line bundle of the moduli stack of stable supercurves.
We begin with the classical case of a degenerating family of usual curve and then consider separately the cases of NS and Ramond nodes.

\subsection{Classical case}\label{KS-class-sec}

Let $\pi:C\to S$ be a family of stable curves over the formal polydisk $S=\Spf(k[\![t,t_1,\ldots,t_d]\!])$ with one special coordinate $t$.
We denote by $S_0\sub S$ the divisor $t=0$ and by $\pi_0:C_0\to S_0$ the induced family over $S_0$.
We assume that there is a section $q:S_0\to C_0$ such that the map $C\to S$ is smooth away from $q(S_0)$, and that the structure sheaf of the
completion of $C$ along $q(S_0)$
is isomorphic to $\OO_S[\![x,y]\!]/(xy-t)$ (so that the section $q$ corresponds to $x=y=t=0$).
Below we will write simply $q$ to denote the relative node $q(S_0)\sub C_0$.

We would like to discuss the Kodaira-Spencer map for such a family.

Let us denote by $\TT_{S,S_0}$ the sheaf of derivations on $S$ preserving the divisor $S_0$.
Let also $\TT_{C,C_0}$ denote the sheaf of vector fields on $C$ preserving $C_0$.
Locally sections of $\TT_{C,C_0}$ correspond to derivations preserving the ideal generated by $t$.
Finally, we denote by $\TT_{C/S}\sub \TT_{C,C_0}$ the relative tangent sheaf.

\begin{lemma}\label{KS-even-lem} (i) There are exact sequences
\begin{equation}\label{tangent-ex-seq}
0\to \TT_{C/S}\to \TT_{C,C_0}\to \pi^*\TT_{S,S_0}\to 0
\end{equation}

\begin{equation}\label{tangent-C0-ex-seq}
0\to \TT_{C_0/S_0}\to \TT_{C_0}\to \pi_0^*\TT_{S_0}\to 0
\end{equation}

\noindent
(ii) For the closed point $x_0\sub S_0$, the natural map
$$\TT_{C_0/S_0}|_{C_{x_0}}\to \TT_{C_{x_0}/k}$$
is an isomorphism.

\noindent
(iii) Let $j:C\setminus\{q\}\to C$ denote the open embedding. Then the natural map $\om_{C/S}\to j_*\om_{C\setminus\{q\}/S}$ is an isomorphism.
The corresponding natural map
$$\Om_{C/S}\to j_*\Om_{C\setminus\{q\}/S}\simeq j_*\om_{C\setminus\{q\}/S}\simeq \om_{C/S}$$
is injective with the cokernel isomorphic to $\OO_q$.
Hence, the dual map
$$\om_{C/S}^{-1}\to \TT_{C/S}$$
is an isomorphism.
Also, the coherent sheaf $\TT_{C,C_0}$ on $C$ is locally free, and the sheaf $\TT_{C_0/S_0}$ on $C_0$ is flat over $S$.

\noindent
(iv) Let us consider the map
$$\TT_C\to \OO_C/(t)=i_*\OO_{C_0},$$
where $i:C_0\to C$ is the natural embedding,
sending $v$ to $v(t)\mod (t)$.
Then its image is $i_*\II_q$, where $\II_q\sub \OO_{C_0}$ is the ideal sheaf of $q$, so we have
an exact sequence
\begin{equation}\label{tangent-C-C0-ex-seq}
0\to \TT_{C,C_0}\to \TT_C\to i_*\II_q\to 0
\end{equation}

\noindent
(v) There is an exact sequence
$$0\to \TT_C/\TT_{C,C_0}\rTo{t} \TT_{C,C_0}/t\TT_{C,C_0}\to i_*\TT_{C_0}$$
\end{lemma}

\begin{proof}
(i) For the first sequence, we have to check that the map $\TT_{C,C_0}\to \pi^*\TT_{S,S_0}$ is surjective. This is clear away from $q$.
It remains to check this in the formal neighborhood of $q$. But we can extend the derivation $t\partial_t$ to the derivation of $k[\![x,y,t]\!]/(xy-t)$ induced by
$x\partial_x+t\partial_t$.

It follows that the composition
$$\TT_{C,C_0}\to \pi^*\TT_{S,S_0}\to i_*\pi_0^*\TT_{S_0}$$
is surjective, which implies surjectivity of the map $\TT_{C_0}\to \pi_0^*\TT_{S_0}$, and hence, exactness of the second sequence.

\noindent
(ii) This follows easily by a local computation near the node. Namely, locally a derivation $v$ in $\TT_{C_0/S_0}$ is described
by a pair of functions $v(x)$ and $v(y)$, satisfying $v(x)y+xv(y)=0$, or equivalently $v(x)\in (x)$, $v(y)\in (y)$, so $\TT_{C_0/S_0}$
is locally isomorphic to $\II_q$, and the assertion follows from this.

\noindent
(iii) The first assertion follows from the fact that $\om_{C/S}$ is locally free.
The injectivity of $\Om_{C/S}\to \om_{C/S}$ only has to be checked in a formal neighborhood of $q$.
Then we can identify $\Om_{C/S}$ with $(\OO dx\oplus \OO dy)/(xdy+ydx)$.
It is easy to see that any element of $\Om_{C/S}$ can be written uniquely as
$$f(t,x,y)dx+ g(y)dy.$$
The map $\Om_{C/S}\to \om_{C/S}$ is given by $dx\mapsto x\cdot \bb$, $dy\mapsto -y\cdot \bb$, where $\bb$ is a generator of $\om_{C/S}$.
Hence, it sends the above element to
$$([f(t,x,y)x-g(y)y)\cdot \bb,$$
which is zero only if $f=g=0$.
The last assertion follows by dualizing the sequence
$$0\to \Om_{C/S}\to \om_{C/S}\to \OO_q\to 0$$
and using the vanishing $\und{\Hom}(\OO_q,\OO_C)=\und{\Ext}^1(\OO_q,\OO_C)=0$.

The fact that $\TT_{C,C_0}$ is locally free follows from the exact sequence \eqref{tangent-ex-seq}.
The sheaf $\TT_{C_0/S_0}$ is locally free away from the node. Thus, to check its flatness it is enough to consider it in the
formal neighborhood of the node. Then it can be identified with the subsheaf of $\OO_{C_0}\oplus \OO_{C_0}$
consisting of $(f,g)$ such that $f\in(x)$ and $g\in (y)$, so it is locally free as $\OO_{S_0}$-module.

\noindent
(iv) Since the map $\pi:C\to S$ is smooth away from $q$, it is enough to check the assertion in the formal neighborhood of the node.
Then we can use $x,y,t_1,\ldots,t_d$ as coordinates on $C$. Since $t=xy$, the map in question $\TT_C\to \OO_C/(t)$ is given by
$$f\partial_x+g\partial_y+\sum f_i\partial_{t_i}\mapsto fy+gx \mod (t),$$
so its image is $\II_q\sub \OO_{C_0}$.

\noindent
(v) This follows easily from the fact that the kernel of the projection $\TT_{C,C_0}\to i_*\TT_{C_0}$ is $t\TT_C\sub \TT_{C,C_0}$.
\end{proof}

It follows that we can rewrite the exact sequence \eqref{tangent-ex-seq} as
\begin{equation}\label{tangent-ex-seq-bis}
0\to \om^{-1}_{C/S}\to \TT_{C,C_0}\to \pi^*\TT_{S,S_0}\to 0.
\end{equation}
Hence, applying the functor $R\pi_*(\cdot)$ and using the isomorphism $\pi_*\OO_C\simeq \OO_S$, we get the induced coboundary map
\begin{equation}\label{KS-even-map}
KS:\TT_{S,S_0}\to R^1\pi_*\om^{-1}_{C/S}
\end{equation}
which we call the {\it Kodaira-Spencer map} for our family, since it restricts to the usual Kodaira-Spencer map over the punctured formal disk.
Note that we also have a similar map coming from the sequence \eqref{tangent-C0-ex-seq},
\begin{equation}\label{KS0-even-map}
KS_0:\TT_{S_0}\to R^1\pi_{0*}\TT_{C_0/S_0}.
\end{equation}
Let $x_0\in S_0$ denote the origin, and let $C_{x_0}$ be the corresponding curve with one node.
Then by Lemma \ref{KS-even-lem}(i), we have $\TT_{C_0/S_0}|_{C_{x_0}}\simeq \TT_{C_{x_0}}$,
so the exact sequence \eqref{tangent-C0-ex-seq} restricts to the standard exact sequence associated with the
embedding $C_{x_0}\hookrightarrow C_0$ (with the trivial normal bundle):
$$0\to \TT_{C_{x_0}}\to \TT_{C_0}|_{C_{x_0}}\to T_{x_0}S_0\ot \OO_{C_{x_0}}\to 0.$$
We can look at the corresponding coboundary map
$$\kappa_{x_0}: T_{x_0}S_0\to H^1(C_{x_0},\TT_{C_{x_0}}).$$

\begin{prop}\label{KS-even-prop}

Assume that the map $\kappa_{x_0}$ is injective.
Then $R^1\pi_*\om^{-1}_{C/S}$ is a vector bundle on $S$
and the map \eqref{KS-even-map} is an embedding of a subbundle.
\end{prop}

\begin{proof}
Since $C_{x_0}$ is a stable curve, we have $H^0(C_{x_0},\om^{-1})=0$, which implies by semicontinuity
that $\pi_*\om^{-1}_{C/S}=0$, and by the base change theorem that
$R^1\pi_*\om^{-1}_{C/S}$ is locally free.
Thus, we have a long exact sequence
$$0\to \pi_*\TT_{C,C_0}\to \TT_{S,S_0}\rTo{KS} R^1\pi_*\om^{-1}_{C/S}\to R^1\pi_*\TT_{C,C_0}\to R^1\pi_*\OO_C\ot \TT_{S,S_0}\to 0.$$
Note that $R^1\pi_*\OO_C$ is locally free of rank $g$. Also $\TT_{C,C_0}$ is a vector bundle on $C$ as follows from the exact sequence
\eqref{tangent-ex-seq}. Thus, if we prove that $H^0(C_{x_0},\TT_{C,C_0}|_{C_{x_0}})=0$ then it would follow that $\pi_*\TT_{C,C_0}=0$ and
$R^1\pi_*\TT_{C,C_0}$ is locally free, and our assertion would follow.

Let us consider the morphism of exact sequences on $C_0$,
\begin{diagram}
0&\rTo{}&\TT_{C/S}|_{C_0}&\rTo{}&\TT_{C,C_0}|_{C_0}&\rTo{}&\pi_0^*\TT_{S,S_0}|_{S_0}&\rTo{}& 0\\
&&\dTo{}&&\dTo{}&&\dTo{}\\
0&\rTo{}&\TT_{C_0/S_0}&\rTo{}&\TT_{C_0}&\rTo{}&\pi_0^*\TT_{S_0}&\rTo{}&0
\end{diagram}
Since the rightmost terms are locally free, the restrictions of these sequences to $C_{x_0}$ are still exact, so we get
a commutative square of the coboundary maps
\begin{diagram}
\TT_{S,S_0}|_{x_0}&\rTo{\kappa'_{x_0}}& H^1(C_{x_0},\om_{C_{x_0}}^{-1})\\
\dTo{r}&&\dTo{}\\
T_{x_0}S_0&\rTo{\kappa_{x_0}}& H^1(C_{x_0},\TT_{C_{x_0}})
\end{diagram}
We claim that the map $\kappa'_{x_0}$ is injective. Indeed, by assumption, $\kappa_{x_0}$ is injective.
Also, the map $r$ is surjective with the $1$-dimensional kernel spanned by $t\partial_t$.
Thus, our claim reduces to the assertion that the restriction of $\kappa'_{x_0}$ to $\ker(r)$ is injective.

To prove the last assertion let us consider the embedding $D\sub S$ of the formal $1$-dimensional disk with the coordinate $t$, so that $D_0=D\cap S_0=\{x_0\}$.
Then we can identify $\ker(r)$ with the image of the natural embedding
$$\TT_{D,x_0}|_{x_0}\to \TT_{S,S_0}|_{x_0}.$$
Applying Lemma \ref{KS-even-lem}(iv)(v) to the induced family of curves $\pi_D:C_D\to D$, we get an exact sequence of sheaves on $C_{x_0}$
$$0\to \II_{q_{x_0}}\rTo{t} \TT_{C_D,C_{x_0}}|_{C_{x_0}}\to \TT_{C_{x_0}}.$$
where $q_{x_0}\in C_{x_0}$ is the node.
Since $H^0(C_{x_0},\TT_{C_{x_0}})=H^0(C_{x_0},\II_{q_{x_0}})=0$, this implies that
$$H^0(C_{x_0},\TT_{C_D,C_{x_0}}|_{C_{x_0}})=0.$$

Therefore, looking at the exact sequence of sheaves on $C_{x_0}$,
$$0\to \om_{C_{x_0}}^{-1}\to \TT_{C_D,C_{x_0}}|_{C_{x_0}}\to (\pi_D^*\TT_{D,x_0})|_{C_{x_0}}\to 0$$
we get that the corresponding coboundary map
\begin{equation}
\label{C-x0-TD-coboundary-eq}
H^0(C_{x_0},\TT_{D,x_0}|_{x_0}\ot\OO_{C_{x_0}})\simeq\TT_{D,x_0}|_{x_0}\to H^1(C_{x_0},\om_{C_{x_0}}^{-1})
\end{equation}
in injective.

Now, restricting to $C_{x_0}$ the natural morphism of exact sequences of vector bundles
\begin{diagram}
0&\rTo{}&\om_{C_D/D}^{-1}&\rTo{}&\TT_{C_D,C_{x_0}}&\rTo{}&\pi_D^*\TT_{D,x_0}&\rTo{}& 0\\
&&\dTo{\sim}&&\dTo{}&&\dTo{}\\
0&\rTo{}&\om_{C/S}^{-1}|_{C_D}&\rTo{}&\TT_{C,C_0}|_{C_D}&\rTo{}&\pi_D^*\TT_{S,S_0}|_D&\rTo{}&0
\end{diagram}
and considering the morphism between the corresponding long exact sequences of cohomology on $C_{x_0}$,
we deduce that the coboundary map \eqref{C-x0-TD-coboundary-eq} is equal to the restriction of $\kappa'_{x_0}$ to $\ker(r)$,
which proves our claim.

Finally, by stability of $C_{x_0}$, we have $H^0(C_{x_0},\om_{C_{x_0}}^{-1})=0$.
Hence, the long exact sequence
$$0\to H^0(C_{x_0},\om_{C_{x_0}}^{-1})\to
H^0(C_{x_0},\TT_{C,C_0}|_{C_{x_0}})\to \TT_{S,S_0}|_{x_0}\rTo{\kappa'_{x_0}}H^1(C_{x_0},\om_{C_{x_0}}^{-1})$$
shows the vanishing of
$H^0(C_{x_0},\TT_{C,C_0}|_{C_{x_0}})$.
\end{proof}

\begin{cor}
In the situation of Proposition \ref{KS-even-prop} assume in addition that the map from $S$ to the moduli space of stable curves induces an isomorphism
on tangent spaces at the origin.
Then the map \eqref{KS-even-map} is an isomorphism.
Hence, the determinant of the usual Kodaira-Spencer map on $S'=S\setminus S_0$,
$$\det KS: \Det \TT_{S'}\to \Det R^1\pi_*\om^{-1}_{C/S}|_{S'}$$
is of the form $u/t$, where $u$ is a unit (with respect to trivializations regular on $S$).
\end{cor}

\begin{proof}
For the second statement, we observe that $\TT_{S,S_0}$ has a basis $t\partial_t,\partial_{t_1},\ldots,\partial{t_d}$ (where $(t,t_1,\ldots,t_d)$ are the
coordinates on $S$. Hence the determinant of the natural morphism
$\TT_{S,S_0}\to \TT_S$ with respect to the standard bases is equal to $t$.
\end{proof}

\subsection{Super case, NS node}\label{super-KS-NS-sec}

First, let us review the Kodaira-Spencer map for a family of smooth supercurves $\pi:X\to S$.
We have the standard exact sequence
$$0\to \TT_{X/S}\to \TT_X\rTo{d\pi} \pi^*\TT_S\to 0.$$
We also have a relative distribution $\DD\sub \TT_{X/S}$.
Let us set

$$\AA_X:=\{v\in \TT_X \ |\ [v,\DD]\sub \DD\}.$$
Then there is an exact sequence (see \cite[Sec.\ 2]{LR})
\begin{equation}\label{A-X-smooth-case-eq}
0\to \AA_{X/S}\to \AA_X\to \pi^{-1}\TT_S\to 0
\end{equation}
where
$$\AA_{X/S}:=\AA_X\cap \TT_{X/S}.
$$
Indeed, let $D$ be a local generator of $\DD$. Since $\DD\sub \TT_{X/S}$,
the condition $[v,D](f)=0$, for $f\in \pi^{-1}\OO_S$, is equivalent to $D(v(f))=0$, i.e., to $v(f)\in \pi^{-1}\OO_S$.
Furthermore, locally any $v\in \pi^{-1}\TT_S$ can be extended to a section of $\AA_X$.

Also, as we have seen before,

$$\AA_{X/S}\simeq \TT_{X/S}/\DD\simeq \DD^2\simeq \om_{X/S}^{-2}.$$
Now the Kodaira-Spencer map is the coboundary map
\begin{equation}
KS: \TT_S\to \pi_*\pi^{-1}\TT_S\to R^1\pi_*\AA_{X/S}\simeq R^1\pi_*\om_{X/S}^{-2}.
\end{equation}

We want to study the behavior of this map near the component of the boundary divisor of the moduli space where a supercurve acquires a NS node.
So let us consider a family of stable supercurves $\pi:X\to S$ over the formal super-polydisk with one special even coordinate $t$.
We denote by $S_0\sub S$ the divisor $t=0$ and by $\pi_0:X_0\to S_0$ the induced family.
We assume that there is a section $q:S_0\to X_0$ such that $q(S_0)$ is the relative node of $X_0$ and the map $X\to S$ is smooth away from $q(S_0)$.
Furthermore, we assume that the structure sheaf of $X$ completed along $q(S_0)$ is generated over $\OO_S$ by even generators $z_1,z_2$ and
odd generators $\th_1,\th_2$, subject to the relations
\begin{equation}\label{NS-node-def-relations}
z_1z_2=-t^2, \ \ z_1\th_2=t\th_1, \ \ z_2\th_1=-t\th_2, \ \ \th_1\th_2=0
\end{equation}
(so that $q(S_0)$ corresponds to $z_1=z_2=0$, $\th_1=\th_2=0$).
In this case the complement to the node is covered by two charts $U_1$ and $U_2$ where $z_i$ is invertible on $U_i$,
and there exist odd sections $s_i$ (resp., even section $s_0$) of $\om_{X/S}$ (defined in the formal neighborhood of the node) such that
\begin{equation}\label{s1-s2-s0-generators}
\begin{array}{l}
s_1=\begin{cases} [dz_1 | d\th_1] & \text{ on } U_1, \\ -\frac{t}{z_2}[dz_2 | d\th_2] & \text{ on } U_2;\end{cases}, \\
s_2=\begin{cases} \frac{t}{z_1}[dz_1 | d\th_1] & \text{ on } U_1, \\ [dz_2 | d\th_2] & \text{ on } U_2;\end{cases}, \\
s_0=\begin{cases} \frac{\th_1}{z_1}[dz_1 | d\th_1] & \text{ on } U_1, \\ -\frac{\th_2}{z_2}[dz_2 | d\th_2] & \text{ on } U_2;\end{cases}
\end{array}
\end{equation}
(see \cite[Sec.\ 2.3]{Deligne}).
We also assume that the relative derivation $\de:\OO_X\to \om_{X/S}$ satisfies
$$\de(z_i)=\th_i s_i, \ \ \de(\th_i)=s_i,$$
for $i=1,2$.

\begin{lemma}\label{NS-om-lem}
In the above situation, in the formal neighborhood of the node, $\om_{X/S}$ is generated as an $\OO_X$-module
by global sections $s_1,s_2$ and $s_0$ subject to defining relations
$$z_1s_2=ts_1, \ \ z_2s_1=-ts_2, \ \ \th_1s_2=\th_2s_1=ts_0, \ \ z_1s_0=\th_1s_1, \ \ z_2s_0=-\th_2s_2, \ \ \th_1s_0=\th_2s_0=0.$$
It has a topological basis over $\OO_S$,
$$z_1^ns_1, \ \ z_2^ns_2,  \ \ z_1^n\th_1s_1, \ \ z_2^n\th_2s_2, \ \ s_0,
$$
where $n\ge 0$.
\end{lemma}

\begin{proof}
It is enough to check this when $S$ is the formal spectrum of $k[\![t]\!]$, in which case this is discussed in {Deligne}.
In more detail, $X$ is split, with $\OO_X=\OO_C\oplus L$, where $C$ is the curve $z_1z_2=-t^2$, and $L$ is the $\OO_C$-module
generated by $\th_1$,$\th_2$ (which is a maximal CM-module over $\OO_C$). Then $\om_X=\om_C\oplus \und{\Hom}(L,\om_C)$,
and we have an isomorphism $\de:L\rTo{\sim}\und{\Hom}(L,\om_C)$.
The section $s_0$ corresponds to the standard generator of $\om_C$, while $s_1$ and $s_2$ are the images of the generators $\th_1,\th_2$ of $L$
under $\de$.
\end{proof}

Note that the line bundle $\om_{X/S}^2:=j_*\om_{U/S}^2$ is generated near the node by the section
\begin{equation}\label{e-sec-eq}
e=\begin{cases} \frac{1}{z_1} [dz_1 |d\th_1]^2 & \text{ on } U_1, \\ \frac{1}{z_2} [dz_2 |d\th_2]^2 & \text{ on } U_2.\end{cases}
\end{equation}

As before, we denote by $\TT_X$ the sheaf of derivations of $\OO_X$
and consider the subsheaf $\AA_X\sub \TT_X$ consisting of $v$ such that $[v,\DD]\sub \DD$ on the smooth locus
of $\pi:X\to S$. We also set $\AA_{X/S}=\AA_X\cap \TT_{X/S}$. We use a similar definition for $\AA_{X_0/S_0}\sub \AA_{X_0}$.

Now similarly to the nodal even case we consider the subsheaf
$\AA_{X,X_0}\sub \AA_X$ consisting of $v$ such that $v(t)\in (t)$. We have a natural projection
$$\AA_{X,X_0}\to \pi^{-1}\TT_{S,S_0}.$$

Note that some of the above sheaves do not have $\OO_X$-module, only the $\pi^{-1}\OO_S$-module structure.
Nevertheless, for a subscheme $S'\sub S$ we have a natural operation of restriction to $\pi^{-1}(S')$: we set for an $\pi^{-1}\OO_S$-module $\FF$,
$$\FF|_{\pi^{-1}(S')}:=\FF\ot_{\pi^{-1}\OO_S}\pi^{-1}\OO_{S'}.$$

Below we will prove an analog of Lemma \ref{KS-even-lem}.
Note that the slight difference from the case of the usual curves is that the sheaves $\AA_X$, $\AA_{X,X_0}$ and $\AA_{X/S}$
are not $\OO_X$-submodules of $\TT_X$. However, as we know from Theorem \ref{inf-aut-thm}, there is an isomorphism
of $\pi^{-1}\OO_S$-modules
$$\AA_{X/S}\simeq \om_{X/S}^{-2}:=j_*\om_{U/S}^{-2},$$
where $j:U=X\setminus\{q\}\to X$ denotes the open embedding and $\om_{X/S}^{-2}$ is a line bundle on $X$ (see Theorem \ref{can-square-thm}).

\begin{lemma}\label{KS-super-NS-lem} (i) There are exact sequences
\begin{equation}\label{tangent-super-ex-seq}
0\to \AA_{X/S}\to \AA_{X,X_0}\to \pi^{-1}\TT_{S,S_0}\to 0
\end{equation}

\begin{equation}\label{tangent-X0-ex-seq}
0\to \AA_{X_0/S_0}\to \AA_{X_0}\to \pi_0^{-1}\TT_{S_0}\to 0
\end{equation}

\noindent
(ii) In the formal neighborhood of a node, derivations in $\AA_{X_0/S_0}$ are in bijection with pairs of functions $a_1,a_2$
in $\OO_{X_0}$ such that $a_i\in (z_i)$: the corresponding derivation $v$ is given by
\begin{equation}\label{v-ai-Di-eq}
v(z_i)=a_i+(-1)^{|v|}\frac{1}{2}D_i(a_i)\th_i, \ \ v(\th_i)=(-1)^{|v|}\frac{1}{2}D_i(a_i)
\end{equation}
for $i=1,2$, where $D_i=\partial_{\th_i}+\th_i\partial_{z_i}$.

For the closed point $s_0\in S_0$, the natural morphism
$$\AA_{X_0/S_0}|_{X_{s_0}}\to \AA_{X_{s_0}/k}$$
is an isomorphism.

\noindent
(iii) One has
$$\AA_{X,X_0}=\AA_X$$

and the natural morphism
$$\AA_{X,X_0}/t\AA_{X,X_0}\to i_*\AA_{X_0}$$
is injective.

\end{lemma}

\begin{proof}
(i) In both cases it is enough to prove surjectivity of the last arrow in the formal neighborhood of $q$.
Let us first check this for $\AA_{X,X_0}\to \pi^{-1}\TT_{S,S_0}$.
It is enough to extend $t\partial_t$ to an even derivation $v$ in the formal neighborhood of $q$ in $X$,
such that $v|_{U_i}$ preserves $\OO_X\cdot (\partial_{\th_i}+\th_i\partial_{z_i})$, for $i=1,2$. For this, we can take $v$ given by
$$v(z_i)=z_i, \ \ v(\th_i)=\frac{1}{2}\th_i, \ \ v(t)=t.$$
This shows that sequence \eqref{tangent-super-ex-seq} is exact.

The composed arrow
$$\AA_{X,X_0}\to \pi^{-1}\TT_{S,S_0}\to i_*\pi_0^{-1}\TT_{S_0}$$
is still surjective and factors through $i_*\AA_{X_0/S_0}$. This implies exactness
of \eqref{tangent-X0-ex-seq}.

\noindent
(ii) Recall that $\OO_{X_0}$ is the quotient
of $\OO_{S_0}[\![z_1,z_2,\th_1,\th_2]\!]$ by the relations
$$z_1z_2=0, \ \ z_1\th_2=0, \ \ z_2\th_1=0, \ \ \th_1\th_2=0.$$
Thus, a derivation $v$ in $\AA_{X_0/S_0}$ is described by the functions
$v(z_1)$, $v(z_2)$, $v(\th_1)$ and $v(\th_2)$. Furthermore, there should exist
a pair of functions $a_1(z_1,\th_1)\in \OO_{X_0}[z_1^{-1}]$, $a_2(z_2,\th_2)\in \OO_{X_0}[z_2^{-1}]$ of the same parity as $v$ such that
$$v(z_i)=a_i+(-1)^{|v|}\frac{1}{2}D_i(a_i)\th_i, \ \ v(\th_i)=(-1)^{|v|}\frac{1}{2}D_i(a_i) \ \text{ in } \OO_{X_0}[z_i^{-1}],$$
for $i=1,2$, where $D_i=\partial_{\th_i}+\th_i\partial_{z_i}$.

The condition $v(z_1z_2)=0$ implies that
$$v(z_i)\in (z_i,\th_i)\sub \OO_{X_0}$$
for $i=1,2$. Hence, $a_i$ is the image of the element $v(z_i)-v(\th_i)\th_i\in (z_i,\th_i)\sub \OO_{X_0}$
under the map $\OO_{X_0}\to \OO_{X_0}[z_i^{-1}]$.
Note that the restriction of the latter map to
$$(z_i,\th_i)=\OO_S[z_i]z_i\oplus \OO_S[z_i]\th_i\sub \OO_{X_0}$$
is an embedding. Thus,
$v(z_i)$ is determined by its image in $\OO_{X_0}[z_i^{-1}]$ and the above formulas for $v(z_i)$ hold in $\OO_{X_0}$ with some
$$a_i=f_i(z_i)z_i+g_i(z_i)\th_i,$$
for $i=1,2$, with $f_i,g_i\in\OO_S[z_i]$.

It follows also that $v(z_1)\th_2=v(z_2)\th_1=0$.
Hence, the conditions $v(z_1\th_2)=v(z_2\th_1)=0$ are equivalent to $v(\th_i)\in (z_i,\th_i)$, for $i=1,2$.
Since $D_i(a_i)\equiv g_i(z_i)\mod (\th_i)$, this is equivalent to $g_i(z_i)\in (z_i)$.
The condition $v(\th_1\th_2)$ is then automatically satisfied.
Note also that the condition $v(\th_i)\in (z_i,\th_i)$ implies that $v(\th_i)$ is determined by its image in $\OO_{X_0}[z_i^{-1}]$.
Thus, the condition on $v$ to define a section of $\AA_{X_0/S_0}$ is that
$a_i\in (z_i)$, for $i=1,2$, which is equivalent to our assertion.

For the last assertion, we note that it is enough to check it in the formal neighborhood of the node. Then the statement follows immediately from
the above explicit description of $\AA_{X_0/S_0}$ and from a similar description of $\AA_{X_{s_0}/k}$.

\noindent
(iii) We claim that in fact any derivation $v$ in $\TT_X$ satisfies $v(t)\in (t,z_1,z_2,\th_1,\th_2)$.
Indeed, we have
$$v(z_1)z_2+z_1v(z_2)=-2tv(t),$$
so $tv(t)\in (z_1,z_2)$.
Let us write $tv(t)=z_1f+z_2f'$, and decompose $f$ and $f'$ with respect to the topological $\OO_S$-basis
$$1, (z_1^n)_{n\ge 1}, (z_2^n)_{n\ge 1}, (z_1^n\th_1)_{n\ge 0}, (z_2^n\th_2)_{n\ge 0}.$$
Then we get the equation of the form
\begin{align*}
&tv(t)=z_1[a_0+\sum_{n\ge 1} a_n z_1^n+\sum_{n\ge 1} b_n z_2^n+\sum_{n\ge 0} c_nz_1^n\th_1+\sum_{n\ge 0}d_nz_2^n\th_2]\\
&+z_2[a'_0+\sum_{n\ge 1} a'_n z_1^n+\sum_{n\ge 1} b'_n z_2^n+\sum_{n\ge 0} c'_nz_1^n\th_1+\sum_{n\ge 0}d'_nz_2^n\th_2]
\end{align*}
The relations imply that the free term of the right-hand side is $-(b_1+a'_1)t^2$, hence the free term of $v(t)$ is divisible by $t$, as claimed.

It follows that for any $v\in \AA_X$, we have $v(t)\in t\pi^{-1}\OO_S$, so $v\in \AA_{X,X_0}$.
For the last assertion we observe that the kernel of
the restriction map
$$\AA_{X,X_0}\to i_*\AA_{X_0}$$
consists of derivations $v$ in $\AA_X$ that are in $t\cdot \TT_X$. Hence, this kernel is $t\AA_X=t\AA_{X,X_0}$.
\end{proof}

Now, as in the even case, we consider the {\it Kodaira-Spencer maps}
associated with sequence \eqref{tangent-super-ex-seq}

\begin{equation}\label{KS-super-NS-map}
KS:\TT_{S,S_0}\to R^1\pi_*\om^{-2}_{X/S},
\end{equation}

Let $s_0\in S_0$ denote the origin, and consider the restriction of the exact sequence \eqref{tangent-X0-ex-seq}
to $X_{s_0}$, which can be identified with
$$0\to \AA_{X_{s_0}}\to \AA_{X_0}|_{X_{s_0}}\to T_{s_0}S_0\ot \und{\C}_{X_{s_0}}\to 0.$$
We have the corresponding coboundary map
$$\kappa_{s_0}:T_{s_0}S_0\to H^1(X_{s_0},\AA_{X_{s_0}}).$$

Now we can prove the following analog of Proposition \ref{KS-even-prop}.

\begin{prop}\label{KS-super-NS-prop}
Assume that the map $\kappa_{x_0}$ is injective.

Then $R^1\pi_*\om^{-2}_{X/S}$ is a vector bundle on $S$
and the map \eqref{KS-super-NS-map} is an embedding of a subbundle.
\end{prop}

\begin{proof} First, let $(C_{s_0},L_{s_0})$ be the usual stable curve with a spin structure, underlying $X_{s_0}$.
We observe that
$$\om^{-2}_{X/S}|_{X_{s_0}}=\om_{C_{s_0}}^{-1}\oplus \om_{C_{s_0}}^{-1}\ot L_{s_0}.$$
Indeed, this follows easily from Theorem \ref{can-square-thm} applied to
the induced family of stable supercurves $X_D\to D$ over the formal disk $D\sub S_0$ with the coordinate $t$.
By stability of $C_{s_0}$, we get
$$H^0(C_{s_0},\om^{-1})=H^0(C_{s_0},\om^{-1}\ot L_{s_0})=0.$$
In other words, we get the vanishing
\begin{equation}\label{X-s0-om-van}
H^0(X_{s_0},\om^{-2}_{X/S}|_{X_{s_0}})=0.
\end{equation}
It follows that
$$\pi_*\AA_{X/S}=\pi_*\om^{-2}_{X/S}=0,$$
and that $R^1\pi_*\om^{-2}_{X/S}$ is a vector bundle on $S$.

Thus, $KS$ is a map of vector bundles on $S$, so it is enough to show that the corresponding map of fibers at $s_0$,
$$KS(s_0):\TT_{S,S_0}|_{s_0}\to (R^1\pi_*\om_{X/S}^{-2})|_{s_0}\simeq H^1(X_{s_0},\om_{X/S}^{-2}|_{X_{s_0}}),$$
is injective.

To this end, let us consider the morphism of exact sequences on $X_0$,
\begin{diagram}
0&\rTo{}&\AA_{X/S}|_{X_0}&\rTo{}&\AA_{X,X_0}|_{X_0}&\rTo{}&\pi_0^{-1}\TT_{S,S_0}|_{S_0}&\rTo{}& 0\\
&&\dTo{}&&\dTo{}&&\dTo{}\\
0&\rTo{}&\AA_{X_0/S_0}&\rTo{}&\AA_{X_0}&\rTo{}&\pi_0^{-1}\TT_{S_0}&\rTo{}&0
\end{diagram}
Since the rightmost terms are locally free, the restrictions of these sequences to $X_{s_0}$ are still exact, so we get
a commutative square of the coboundary maps
\begin{diagram}
\TT_{S,S_0}|_{s_0}&\rTo{KS(s_0)}& H^1(X_{s_0},\om_{X/S}^{-2}|_{X_{s_0}})\\
\dTo{r}&&\dTo{}\\
T_{s_0}S_0&\rTo{\kappa_{s_0}}& H^1(X_{x_0},\AA_{X_{s_0}})
\end{diagram}
Since $\kappa_{s_0}$ is injective, arguing as before, we see that it is enough to check
that the restriction of $KS(s_0)$ to the $1$-dimensional subspace
$$\TT_{D,s_0}|_{s_0}\hra \TT_{S,S_0}|_{s_0}$$
is injective.

Considering the induced family $\pi_D:X_D\to D$ and applying Lemma \ref{KS-super-NS-lem}(iv), we get
an inclusion of sheaves on $X_{x_0}$,
$$\AA_{X_D,X_{s_0}}|_{X_{s_0}}\hra \AA_{X_{s_0}}.$$
We know that $H^0(X_{s_0},\AA_{X_{s_0}})=0$ (see Prop.\ \ref{no-inf-aut-prop}(ii)), so we get that
$$H^0(X_{s_0},\AA_{X_D,X_{s_0}}|_{X_{s_0}})=0.$$

Next, restricting the sequence \eqref{tangent-super-ex-seq} for the family $X_D/D$ to $X_{s_0}$, we get an exact sequence
$$0\to \om_{X_D}^{-2}|_{X_{s_0}}\to \AA_{X_D,X_{s_0}}|_{X_{s_0}}\to (\pi_D^{-1}\TT_{D,s_0})|_{X_{s_0}}\to 0.$$
From the above vanishing we get that the corresponding coboundary map
\begin{equation}
\label{X-s0-TD-super-coboundary-eq}
H^0(X_{s_0},\TT_{D,s_0}|_{s_0}\ot\OO_{X_{s_0}})\simeq\TT_{D,s_0}|_{s_0}\to H^1(X_{s_0},\om_{X_D}^{-2}|_{X_{s_0}})
\end{equation}
is injective.

Now, restricting to $X_{s_0}$ the natural morphism of exact sequences
\begin{diagram}
0&\rTo{}&\om_{X_D/D}^{-2}&\rTo{}&\AA_{X_D,X_{s_0}}&\rTo{}&\pi_D^{-1}\TT_{D,s_0}&\rTo{}& 0\\
&&\dTo{\sim}&&\dTo{}&&\dTo{}\\
0&\rTo{}&\om_{X/S}^{-2}|_{X_D}&\rTo{}&\AA_{X,X_0}|_{X_D}&\rTo{}&\pi_D^{-1}\TT_{S,S_0}|_D&\rTo{}&0
\end{diagram}
and considering the morphism between the corresponding long exact sequences of cohomology on $X_{s_0}$,
we deduce that the coboundary map \eqref{X-s0-TD-super-coboundary-eq} is equal to the restriction of $KS(s_0)$ to $\ker(r)$,
which proves our claim.

\end{proof}

Note that the Berezinian of the natural morphism $\TT_{S,S_0}\to \TT_S$ is equal to $t$ for an appropriate choice of bases.

\begin{cor}\label{KS-super-NS-node-cor}
Assume that the map $\kappa_{x_0}$ is an embedding and the superdimension of $S$ is $3g-3|2g-2$.
Then the map \eqref{KS-super-NS-map} is an isomorphism. Hence, the Berezinian of the usual Kodaira-Spencer map on $S'=S\setminus S_0$,
$$\ber(KS):\Ber(\TT_{S'})\to \Ber R^1\pi_*\om_{X'/S'}^{-2}$$
is of the form $u/t$, where $u$ is invertible on $S$ (with respect to trivializations regular on $S$).
\end{cor}

\subsection{Super case, Ramond node}\label{super-KS-R-sec}

Now we consider a family $X\to S$ over the formal polydisk with Ramond node $q:S_0\to X_0=\pi^{-1}(S_0)$ over the divisor $S_0=(t=0)$,
where the completion of $\OO_X$ along $q(S_0)$ is generated over $\OO_S$ by generators $z_1,z_2,\th$ subject to the single relation
$$z_1z_2=t.$$
The distribution is generated by $\partial_{\th}+\th z_i\partial_{z_i}$ over $z_i\neq 0$.

Note that in this case the relative dualizing sheaf $\om_{X/S}$ is a line bundle on $X$.

\begin{lemma}\label{KS-super-R-lem} (i) The sequences
\eqref{tangent-super-ex-seq} and \eqref{tangent-X0-ex-seq} are still exact.

\noindent
(ii) In the formal neighborhood of a node, even (resp., odd) derivations in $\AA_{X_0/S_0}$ are in bijection with pairs of even (resp., odd) functions
$a_i=f_i(z_i)+g_i(z_i)\th\in \OO_S[z_i,\th_i]$, $i=1,2$,
such that $g_1(0)=g_2(0)$.
The corresponding derivation $v$ is given by
$$v(z_i)=a_iz_i+(-1)^{|v|}\frac{1}{2}D_i(a_i)z_i\th, \ \text{ for } i=1,2,$$
$$v(\th)=(-1)^{|v|}\frac{1}{2}[(-1)^{|v|}(g_1+g_2-g_2(0))+\th z_1\partial_{z_1}(f_1)+\th z_2\partial_{z_2}(f_2)].$$
where $D_i=\partial_{\th}+\th z_i\partial_{z_i}$.

For the closed point $s_0\in S_0$, the natural morphism
$$\AA_{X_0/S_0}|_{X_{s_0}}\to \AA_{X_{s_0}/k}$$
is an isomorphism.

\noindent
(iii) One has
$$\AA_{X,X_0}=\AA_X$$

and the natural morphism
$$\AA_{X,X_0}/t\AA_{X,X_0}\to i_*\AA_{X_0}$$
is injective.

\end{lemma}

\begin{proof}
(i) To show surjectivity of the morphism $\AA_{X,X_0}\to \pi^{-1}\TT_{S,S_0}$
we observe that the derivation $v\in \AA_{X,X_0}$, given by
$$v(t)=2t, \ \ v(z_i)=z_i, \ \ v(\th)=0,$$
extends $2t\partial_t$.

\noindent
(ii)
Here is the description of $\AA_{X_0/S_0}$.
The condition $v(z_1)z_2+v(z_2)z_1=0$ implies that $v(z_i)\in (z_i)$, in particular $v(z_i)$ is determined by its image in $\OO_{X_0}[z_i^{-1}]$.
Also, we should have $a_i\in \OO_S[z_i,z_i^{-1},\th]$ such that
$$v(z_i)=a_iz_i+(-1)^{|v|}\frac{1}{2}D_i(a_i)z_i\th, \ \ v(\th)=(-1)^{|v|}\frac{1}{2} D_i(a_i) \ \text{ in } \OO_{X_0}[z_i^{-1}].$$
Let us write $a_i=f_i(z_i)+g_i(z_i)\th$. Then $a_1$ and $a_2$ could be arbitrary elements of $\OO_S[z_1,\th]$ and $\OO_S[z_2,\th]$ (of the same parity) such that
$g_1(0)=g_2(0)$.

\noindent
(iii) The proof is similar to that of Lemma \ref{KS-super-NS-lem}(iv):
we use the condition $v(t)\in (z_1,z_2)\sub \OO_X$ and the
topological $\OO_S$-basis of $\OO_X$
$$1, (z_1^n)_{n\ge 1}, (z_2^n)_{n\ge 1}, (z_1^n\th)_{n\ge 0}, (z_2^n\th)_{n\ge 0}.$$
\end{proof}

Using the same arguments as before we derive the following assertion.

\begin{prop}\label{KS-super-R-prop}
The statements of Proposition \ref{KS-super-NS-prop} and Corollary \ref{KS-super-NS-node-cor} hold
in the case of a Ramond degeneration as well.
\end{prop}

\subsection{Kodaira-Spencer in the presence of NS and Ramond punctures}\label{Kodaira-punct-sec}

Everything in Sec.\ \ref{super-KS-NS-sec} and \ref{super-KS-R-sec} has an analog in the case of families of stable supercurves with punctures.
Namely, for such a family $(X,P_\bullet,R_\bullet)/S$, we should replace the sheaves $\AA_X$ and $\AA_{X/S}$
with their intersections $\AA_{X,P,R}$ and $\AA_{(X,P,R)/S}$
with the subsheaf $\TT_{X,P,R}\sub \TT_X$ of derivations preserving all the punctures (i.e., the corresponding ideals in $\OO_X$).
This will not change the local picture near the nodes, however, we need to make some changes in the global statements.

Let us assume that we have a family $(X,P,R)$ of stable supercurves with punctures over $S$, acquiring a single node over $S_0=(t=0)\sub S$,
so that forgetting the punctures we are in the situation of either Sec.\ \ref{super-KS-NS-sec} or Sec.\ \ref{super-KS-R-sec}.
We will still have the equality $\AA_{X,P,R}=\AA_{X,X_0,P,R}$ as in Lemma \ref{KS-super-NS-lem}, and the analogs of sequences \eqref{tangent-super-ex-seq}
and \eqref{tangent-X0-ex-seq} are
$$0\to \AA_{(X,P,R)/S}\to \AA_{X,P,R}\to \pi^{-1}\TT_{S,S_0}\to 0,$$
$$0\to \AA_{(X_0,P_0,R_0)/S}\to \AA_{X_0,P_0,R_0}\to \pi_0^{-1}\TT_{S_0}\to 0.$$
Indeed, the only extra statement is the surjectivity of the right arrows near the punctures, which follows from the standard local description of the punctures
(see Sec.\ \ref{local-descr-sec}).

Recall also that by Theorem \ref{inf-aut-thm}, we have an isomorphism
$$\AA_{(X,P_\bullet,R_\bullet)/S}\simeq \LL(X,P_\bullet,R_\bullet):=\om_{X/S}^{-2}(-\sum_{i\in I} D_i-2\sum_{j\in J} R_j),$$
where $D_i\sub X$ are the divisors associated with the NS nodes $P_i\sub X$, and $\om_{X/S}^{-2}$ is defined as in Theorem \ref{can-square-thm}.

The analog of Corollary \ref{KS-super-NS-node-cor} states that under the assumption that the map
$$\kappa_{x_0}:T_{s_0}S_0\to H^1(X_{s_0}, \AA_{X_{s_0},P_{s_0},R_{s_0}})$$
is injective and the superdimension of $S$ is $3g-3|2g-2$, the Kodaira-Spencer map induces an isomorphism of line bundles on $S$,
$$\Ber(\TT_S)\rTo{\sim} \Ber(R^1\pi_*\LL(X,P_\bullet,R_\bullet))(S_0).$$

\section{The boundary divisor}\label{boundary-str-sec}

In this section we discuss the definition of the boundary of our compactification as an effective Cartier divisor. We also study the NS and the Ramond components
of the boundary divisor.

\subsection{Modified sheaf of differentials}

Let $\pi:X\to S$ be a family of stable supercurves such that the map from $S$ to the deformation space of each node is smooth.
Let $j:U\to X$ be the embedding of the smooth locus. We want to study the sheaf $j_*\Om_{U/S}$ and the map
$$\Om_{X/S}\to j_*\Om_{U/S}.$$

Since $U/S$ is a smooth supercurve, we have an exact sequence
$$0\to \om^2_{U/S}\rTo{\kappa} \Om_{U/S}\rTo{\de} \om_{U/S}\to 0$$
where in standard coordinates
$$\de(dz)=\th [dz|d\th], \ \ \de(d\th)=[dz|d\th],$$
$$\kappa([dz|d\th]^{\ot 2})=dz-\th d\th.$$
Applying the functor $j_*$, we get an exact sequence
$$0\to \om^2_{X/S}\rTo{\kappa} j_*\Om_{U/S}\rTo{j_*\de} \om_{X/S},$$
where $\om^2_{X/S}=j_*\om_{U/S}$ is a line bundle on $X$ (see Theorem \ref{can-square-thm}).

\begin{lemma} The map $j_*\de$ is surjective, so we have an exact sequence
\begin{equation}\label{j-Om-ex-seq}
0\to \om^2_{X/S}\rTo{\kappa} j_*\Om_{U/S}\rTo{j_*\de} \om_{X/S}\to 0
\end{equation}
\end{lemma}

\begin{proof} We know that $j_*\de$ is surjective over $U$, so it is enough to check that it is surjective near the nodes.

Near a Ramond node, the map $\de:\Om_{X/S}\to \om_{X/S}=j_*\om_{U/S}$ is surjective. Since it factors through
$j_*\Om_{U/S}$, we see that $j_*\de$ is surjective near a Ramond node.

It is well known that to check surjectivity near an NS-node, we can pass to completions.
Hence, it remains to make a computation for the standard degeneration of the NS-node.
Recall the generators $s_1,s_2,s_0$ of $\om_{X/S}$ (see Lemma \ref{NS-om-lem}).
We know that $s_1$ and $s_2$ are in the image of $\de:\Om_{X/S}\to \om_{X/S}$.
On the other hand, the sections $dz_1/z_1$ and $-dz_2/z_2$ of $\Om_{U/S}$ over $z_1\neq 0$ and $z_2\neq 0$ glue into a global section of
$j_*\Om_{U/S}$, which maps to $s_0$ under $j_*\de$.
\end{proof}

Now let us consider a family of stable supercurves $X/S$ over the formal super-polydisk with a special even coordinate $t$ such that over $S_0=(t=0)\sub S$,
we have a relative NS node $q:S_0\to X_0=S_0\times S X$, so that the completion of $\OO_X$ along $q(S_0)$ is given by the standard generators and relations
\eqref{NS-node-def-relations}.

\begin{lemma}\label{differential-generators-lem}
(i) In the above situation the sheaf $j_*\Om_{U/S}$ is generated in a formal neighborhood of $q(S_0)$, as an $\OO_X$-module, by global sections
$$e=\frac{dz_1}{z_1}=-\frac{dz_2}{z_2}, \ \ d\th_1, \ \ d\th_2, \ \ f=\frac{\th_1d\th_1}{z_1}=-\frac{\th_2d\th_2}{z_2},$$
with defining relations
$$td\th_1-z_1d\th_2=t\th_1 e, \ \ z_2d\th_1+td\th_2=-t\th_2e,$$
$$tf=\th_2d\th_1=\th_1d\th_2, \ \ z_1f=\th_1d\th_1, \ \ z_2f=-\th_2d\th_2, \ \ \th_1f=\th_2f=0.$$

\noindent
(ii) We have an exact sequence on $X_0$,
$$0\to \KK\to \Om_{X_0/S_0}\to j_*\Om_{U/S}|_{X_0}\to \CC_0\to 0$$
where $\KK$ has an $\OO_{S_0}$-basis
\begin{equation}\label{KK-basis-eq}
z_1dz_2=-z_2dz_1, \ \ z_1d\th_2=-\th_2dz_1, \ \ z_2d\th_1=-\th_1dz_2, \ \ \th_1d\th_2=\th_2d\th_1,
\end{equation}
and $\CC_0$ has an $\OO_{S_0}$-basis the images of
\begin{equation}\label{CC-basis-eq}
e, \ \  f, \ \ \th_1e, \ \ \th_2e.
\end{equation}
Furthermore, the map of $\OO_{S_0}$-modules $\pi_*\KK\to \pi_*\Om_{X_0/S_0}$ is an embedding of a direct summand.
\end{lemma}

\begin{proof}
(i) We can think of sections of $j_*\Om_{U/S}$ as compatible sections of $\Om_{U_1/S}$ and $\Om_{U_2/S}$, where $U_i$ is the open subset
where $z_i$ is invertible. Thus, $e$ is a well defined section of  $j_*\Om_{U/S}$.
Using relations \eqref{NS-node-def-relations}, we get
$$z_2d\th_1=-td\th_2-\th_1dz_2,$$
so over $U_1\cap U_2$,
$$\frac{\th_1}{z_1}d\th_1=\frac{\th_2}{t}(-\frac{t}{z_2}d\th_2)=-\frac{\th_2}{z_2}d\th_2,$$
so $f$ is a well defined section of $j_*\Om_{U/S}$.

It is straightforward to check the relations between $e$, $f$, $d\th_1$ and $d\th_2$ (by inverting $z_1$ or $z_2$).
The map $\kappa:\om^2_{X/S}\to j_*\Om_{U/S}$ sends a generator to $e-f$. Thus, if we denote by $\FF$ the $\OO_X$-module given by our generators and relations
then we have a commutative square
\begin{diagram}
\OO_X &\rTo{e-f}& \FF\\
\dTo{\sim}&&\dTo{}\\
\om^2_{X/S}&\rTo{}&j_*\Om_{U/S}
\end{diagram}
One can check using Gr\"obner basis technique that
$$z_1^ne, \ \ z_1^n\th_1e, \ \  z_1^nd\th_1, \ \ z_1^n\th_1d\th_1, \ \ z_2^ne, \ \ z_2^n\th_2e, \ \ z_2^nd\th_2, \ \ z_2^n\th_2d\th_2, \ \ f,
$$
where $n\ge 0$, is a topological $\OO_S$-basis of $\FF$. This easily implies
that the map $\OO_X\rTo{e-f}\FF$ is injective (by looking at the coefficients of $e$).

The quotient $\FF/\OO_X(e-f)$ is generated by $e,d\th_1,d\th_2$ subject to relations
$$td\th_1-z_1d\th_2=0, \ \ z_2d\th_1+td\th_2=0,$$
$$te=\th_2d\th_1=\th_1d\th_2, \ \ z_1e=\th_1d\th_1, \ \ z_2e=-\th_2d\th_2, \ \ \th_1e=\th_2e=0.$$
Note that
$$\de(d\th_1)=s_1,\ \ \de(d\th_2)=s_2, \ \ \de(e)=s_0.$$
Comparing these with generators and relations of $\om_{X/S}$ (see Lemma \ref{NS-om-lem})
we deduce that the map $\FF/\OO_X(e-f)\to \om_{X/S}$ is an isomorphism.
Now the assertion follows from the exact sequence \eqref{j-Om-ex-seq}.

\noindent
(ii) From (i) we see that $j_*\Om_{U/S}$ is a coherent sheaf (since this can be checked after passing to completion at the node).
Hence, the kernel $\KK$ and the cokernel $\CC_0$ of the map $\Om_{X_0/S_0}\to j_*\Om_{U/S}|_{X_0}$ are coherent sheaves supported at the node.
Also, we get an explicit description of the completion of $j_*\Om_{U/S}|_{X_0}$ at the node by generators and relations.
Taking the quotient by the image of $\Om_{X_0/S}$,
we get that $\CC_0$ (which coincides with its completion at the node) is generated by the images $(\ov{e},\ov{f})$of $(e, f)$, with the defining relations
$$z_1\ov{e}=0, \ \ z_2\ov{e}=0, \ \ z_1\ov{f}=0, \ \ z_2\ov{f}=0, \ \ \th_1\ov{f}=0, \ \ \th_2\ov{f}=0.$$
This implies the assertion that the elements \eqref{CC-basis-eq} constitute an $\OO_{S_0}$-basis of $\CC$.

The kernel $\KK$ coincides with the kernel of the restriction $\Om_{X_0/S_0}\to \Om_{X_1/S_0}\oplus \Om_{X_2/S_0}$. The $4$ elements \eqref{KK-basis-eq}
can be extended to an $\OO_{S_0}$-basis of $\Om_{X_0/S_0}$: we have to add
$$z_i^ndz_i, \ z_i^n\th_idz_i, \ z_i^nd\th_i, \ z_i^n\th_id\th_i,$$
where $i=1,2$, $n\ge 0$. Since the latter elements project to independent elements of $\Om_{X_1/S_0}\oplus \Om_{X_2/S_0}$,
the assertion follows.
\end{proof}

Next let us consider the case of a Ramond node.
We assume that $X/S$ is a family of stable supercurves over the formal super-polydisk with a special even coordinate $t$ such that over $S_0=(t=0)\sub S$,
we have a relative Ramond node $q:S_0\to X_0=S_0\times S X$, so that the completion of $\OO_X$ along $q(S_0)$ is given by the standard generators $z_1,z_2,\th$ subject
to $z_1z_2=t$, and with the distribution generated by $\partial_{\th}+\th z_i\partial_{z_i}$ over $z_i\neq 0$.

\begin{lemma}\label{differential-generators-R-lem}
(i) In the above situation the sheaf $j_*\Om_{U/S}$ is freely generated in a formal neighborhood of $q(S_0)$, as an $\OO_X$-module, by global sections
$$e=\frac{dz_1}{z_1}=-\frac{dz_2}{z_2}, \ \ d\th.$$

\noindent
(ii) We have an exact sequence on $X_0$,
$$0\to \KK\to \Om_{X_0/S_0}\to j_*\Om_{U/S}|_{X_0}\to \CC_0\to 0$$
where $\KK\simeq q_*\OO_{S_0}$ is generated by $z_1dz_2=-z_2dz_1$
and $\CC_0\simeq q_*\OO_{S_0}$ is generated by the image of $e$.
Furthermore, the map of $\OO_{S_0}$-modules $\pi_*\KK\to \pi_*\Om_{X_0/S_0}$ is an embedding of a direct summand.
\end{lemma}

\begin{proof} The proof is similar to that of Lemma \ref{differential-generators-lem} but is much easier since both $\om_{X/S}$ and $\om_{X/S}^2$ are locally free near
the Ramond node. Namely, $\om_{X/S}$ is freely generated by the section glued from $\frac{1}{z_1}[dz_1|d\th]$ and $-\frac{1}{z_2}[dz_2|d\th]$,
The map $\kappa$ sends a generator of $\om_{X/S}^2$ to $e-\th d\th$.
Since $\de$ sends $d\th$ to a generator of $\om_{X/S}$, we deduce that $e$ and $d\th$ freely generate $j_*\Om_{U/S}$ over $\OO_X$.

The map $\Om_{X_0/S_0}\to j_*\Om_{U/S}|_{X_0}$ sends $dz_1$ to $z_1e$, $dz_2$
to $-z_2e$, and $d\th$ to $d\th$. This easily implies that its kernel is generated by $z_2dz_1=-z_1dz_2$, while its cokernel is generate by the image of $e$.
\end{proof}

\subsection{Boundary Cartier divisor}\label{boundary-equation-sec}

The boundary divisor of the compactified moduli superspace has codimension $1|0$, however, since the supermoduli space is not reduced,
the structure of the Cartier divisor on the boundary is not automatically given. We claim however that there is a natural such structure.

Let $\pi:X\to S$ be a family of stable supercurves, inducing surjection to tangent space of deformations of each node???.
For this we are going to study the natural $2$-term complex
$$[\Om_{X/S}\to j_*\Om_{U/S}]$$
placed in degrees $-1$ and $0$.
Note that it is an acyclic over $U$. Furthermore, both terms are flat over $S$, so
the line bundle $\Ber\pi_*[\Om_{X/S}\to j_*\Om_{U/S}]$ is well defined. Similarly
to the theory in the even case (see \cite{KM}), we can define a canonical trivialization of this line bundle away from the locus of nodal supercurves.

\begin{prop}\label{Ber-boundary-prop}
The canonical trivialization $c$ of $\BB:=\Ber\pi_*[\Om_{X/S}\to j_*\Om_{U/S}]$ away from the nodal locus extends to a regular section of $\BB$ on $S$ that has form $t\cdot b$ near
a point corresponding to a stable supercurve with one NS or Ramond node, for some trivializing local section $b$ of $\BB$ and some local function $t$ whose reduction
modulo nilpotents gives an equation of the corresponding reduced divisor.

\end{prop}

\begin{proof}
It is enough to replace $S$ be the formal neighborhood of a point corresponding to a stable supercurve with one node.

Let us first consider the case of an NS node.
We can choose a sufficiently positive divisor $D$ (disjoint from the node) such that $\pi_*[\Om_{X/S}\to j_*\Om_{U/S}]$ will be represented by
the compex of supervector bundles
$$\pi_*(\Om_{X/S}(D))\rTo{\iota} \pi_*(j_*\Om_{U/S}(D)).$$
To understand this map, we first restrict it to $X_0$.
We have an exact sequence on $S$,
$$0\to \pi_*\KK\to \pi_*(\Om_{X_0/S_0}(D))\to \pi_*(j_*\Om_{U/S}(D)|_{S_0})\to \pi_*\CC_0\to 0,$$
where both $\pi_*\KK$ and $\pi_*\CC_0$ are locally free of rank $2|2$.
Furthermore, $\pi_*\KK$ embeds into $\pi_*(\Om_{X_0/S_0}(D))$ as a subbundle, and we have an $\OO_{S_0}$-basis \eqref{KK-basis-eq} of
$\pi_*\KK$. Note that the elements of this basis are defined in terms of the coordinates which only exist locally near the node. However,
they are killed by the ideal $I_q$ of the node, so they can be viewed as sections of $\pi_*\Om_{X_0/S_0}(D)$.

We can choose liftings of the $\OO_{S_0}$-basis \eqref{KK-basis-eq} in $\pi_*\KK$ to $\OO_S$-independent sections $b_1,b_2,b_3,b_4$ in $\pi_*(\Om_{X/S}(D))$, so
that
$$b_1\equiv z_1dz_2 \mod(t), \ b_2\equiv \th_2dz_1 \mod(t), \ b_3\equiv \th_1dz_2 \mod(t), \ b_4\equiv \th_2d\th_1 \mod(t).$$
Then we can extend it to a basis $(b_1,\ldots,b_n)$ of $\pi_*(\Om_{X/S}(D))$ so that the elements $(\iota(b_5),\ldots,\iota(b_n))$
form a basis of a free $\OO_S$-submodule in $\pi_*(j_*\Om_{U/S}(D))$, which projects modulo $(t)$ to a basis of the image of
$\pi_*\Om_{X_0/S_0}(D)$.

Next, we adjust our choice of $b_1$ so that
$$b_1\equiv z_1dz_2 \mod(t^2).$$

Here the right-hand side can be viewed as a section of
$$\pi_*(\Om_{X/S}(D))/(t^2)=\pi_*(\Om_{X_0^{(1)}/S_0^{(1)}}(D)),$$
where we consider the base change $X_0^{(1)}\to S_0^{(1)}$, where $S_0^{(1)}\sub S$ is given by the ideal $t^2$.
Namely, we observe that near the node one has $I_q^2\sub (z_1,z_2)$, which implies that $z_1dz_2=-z_2dz_1$ is killed by $I_q^2$ on $X_0^{(1)}$
(due to the relation $z_1z_2=-t^2$).
Thus, we can lift $z_1dz_2 \mod(t^2)$ to a section $b_1\in \pi_*(\Om_{X/S}(D))$.

Now let us analyse the images of $b_1,\ldots,b_4$ under $\iota$.
We have $\iota(b_i)=tc_i$ for some $c_i\in \pi_*(j_*\Om_{U/S}(D))$, for $i=2,3,4$.
In the case of $b_1$ we know that $\iota(b_1)=t^2c_1$ for some $c_1\in \pi_*(j_*\Om_{U/S}(D))$.

Furthermore, the restriction of $\iota(b_i)$ to a formal neighborhood $\hat{X}$ of the node is the image
of the restriction of $b_i$ to this formal neighborhood under the similar map for $\hat{X}$.
Since $b_1$ restricts to $z_1dz_2+t^2x$, it maps to
$$-z_1z_2e+t^2x=t^2(e+x),$$
where $x$ comes from $\Om_{\hat{X}/S}$.
Hence,
$$c_1\equiv e \mod \Om_{\hat{X}/S}.$$
Similarly, we compute
$$\th_2dz_1=\th_2z_1e=t\th_1e,$$
$$\th_1dz_2=-\th_1z_2e=t\th_2e,$$
$$\th_2d\th_1=tf.$$
Hence, we get
$$c_2\equiv \th_1e\mod \Om_{\hat{X}/S}, \ \ c_3\equiv \th_2e \mod \Om_{\hat{X}/S}, \ \ c_4\equiv f \mod \Om_{\hat{X}/S}.$$
It follows that $c_1,c_2,c_3,c_4$ project to an $\OO_S$-basis of $\pi_*\CC_0$.
Hence, $c_1,c_2,c_3,c_4$ are linearly indpendent over $\OO_S$, and
can be extended to an $\OO_S$-basis $(c_1,\ldots,c_n)$ of $\pi_*(j_*\Om_{U/S}(D))$ by $c_i=\iota(b_i)$ for $i>4$.
Computing the Berezinian in this basis we get $t$.

The case of Ramond node is considered in a similar way using Lemma \ref{differential-generators-R-lem}.
In this case we can choose bases $(b_1,\ldots,b_n)$ of $\pi_*(\Om_{X/S}(D))$ and $(c_1,\ldots,c_n)$ of $\pi_*(j_*\Om_{U/S}(D))$, such that
$b_1\equiv z_1dz_2 \mod (t^2)$, $\iota(b_1)=-tc_1$ (since $z_1dz_2=-z_1z_2e=-te$ in $j_*\Om_{U/S}(D)$), and $\iota(b_i)=c_i$ for $i>1$.
This shows that the Berezinian is $t$.

\end{proof}

The above Proposition gives a natural definition of the boundary divisor in the moduli space $\ov{\SS}$ as an effective Cartier divisor.
More precisely, let $k:\SS'\hra \ov{\SS}$ be the complement to the locus of stable supercurves with more than $1$ node. Since
the even codimension of the latter locus is $>1$ and $\ov{\SS}$ is smooth, for any vector bundle $\VV$ over $\ov{\SS}$, the natural map
$\VV\to k_*k^*\VV$ is an isomorphism.
Let us consider the line bundle
$$\LL:=\Ber\pi_*[\Om_{X/\ov{\SS}}\to j_*\Om_{U/\ov{\SS}}]$$
over $\ov{\SS}$.
By Proposition \ref{Ber-boundary-prop}, the canonical trivialization $c$ of $\LL$ over the smooth locus gives a regular global section of $k^*\LL$
and hence of $\LL\simeq k_*k^*\LL$.
We define the effective Cartier divisor $\De$ to be the vanishing divisor of this global section.
Note that by definition, we have an isomorphism
\begin{equation}\label{boundary-line-bundle}
\OO(\De)\simeq \LL\simeq \Ber R\pi_*(j_*\Om_{U/S})\ot \Ber^{-1} R\pi_*(\Om_{X/S}).
\end{equation}

The proof of Proposition \ref{Ber-boundary-prop} also yields the following characterization of the divisor $\De$.

\begin{lemma}\label{Cartier-char-lem}
The effective Cartier divisor $\De$ is a unique Cartier divisor supported on the locus of nodal supercurves with the following two properties:
\begin{itemize}
\item for any stable supercurve $X_0$ with a single NS node $q$ there exists an \'etale neighborhood $S$ of the corresponding
point $[X_0]$ in the moduli space and an \'etale neighborhood $X_S$ of
$q$ in the family $X\to S$ induced by the universal family such that $\OO_{X_S}$ is generated over $\OO_S$ by $z_1,z_2,\th_1,\th_2$ subject to relations
\eqref{NS-node-def-relations}, where $t$ is a local equation of $\De$ on $S$;
\item for any stable supercurve $X_0$ with a single Ramond node $q$ there exists an \'etale neighborhood $S$ of $[X_0]$ in the moduli space and an \'etale neighborhood $X_S$ of
$q$ in the family $X\to S$ induced by the universal family such that $\OO_{X_S}$ is generated over $\OO_S$ by $z_1,z_2,\th$ subject to the relation
$z_1z_2=t$, where $t$ is a local equation of $\De$ on $S$;
\end{itemize}
\end{lemma}

\begin{proof} These conditions clearly characterize $\De\cap \SS'$. To show uniqueness of an extension to $\ov{\SS}$, we observe that if $\II\sub \OO_{\ov{\SS}}$ is
an invertible ideal sheaf then $\II$ is identified with $k_*k^*\II\sub k_*\OO_{\SS'}=\OO_{\ov{\SS}}$.
\end{proof}

\subsection{Another definition of the boundary divisor and the normal crossing property}\label{normal-cross-sec}

Let $X/\ov{\SS}$ be the universal stable supercurve.

\begin{lemma}
The natural morphism $\AA_X\to \pi^*\TT_{\ov{\SS}}$ of sheaves on $X$
factors through a morphism $\AA_X\to \pi^{-1}\TT_{\ov{\SS}}$.
\end{lemma}

\begin{proof}
This follows the fact that $\AA_X=j_*\AA_U$, where $U\sub X$ is the complement to the nodes (see the proof of Theorem \ref{inf-aut-thm}),
and from the corresponding statement for the smooth locus $U\to \ov{\SS}$ (see Eq. \eqref{A-X-smooth-case-eq}).
\end{proof}

Let us consider the natural morphism
\begin{equation}\label{boundary-eq}
\pi_*(\AA_X/\AA_{X/\ov{\SS}})\to \TT_{\ov{\SS}}
\end{equation}
induced by the map $\AA_X\to \pi^{-1}\TT_{\ov{\SS}}$.

\begin{prop} The $\OO_{\ov{\SS}}$-module $\pi_*(\AA_X/\AA_{X/\ov{\SS}})$ is locally free of rank $(3g-3|2g-2)$ over $\ov{\SS}$, and the Cartier divisor
associated with the Berezinian of the map \eqref{boundary-eq} coincides with $\De$. Furthermore, $\De$ is a normal crossing divisor.
\end{prop}

\begin{proof}

Note that \eqref{boundary-eq} is an isomorphism over the locus of smooth supercurves.
By Lemma \ref{KS-super-NS-lem}, in an \'etale neighborhood $S$ of a point $[X_0]$, where $X_0$ has a single NS node, the image of the morphism
$\AA_X\to \pi^{-1}\TT_S$ is $\pi^{-1}\TT_{S,S_0}$, where $S_0$ is the divisor $(t)$.
Hence, in this case \eqref{boundary-eq} is the embedding $\TT_{B,B_0}\hra \TT_B$, and the Berezinian of this morphism is $t$.
By Lemma \ref{KS-super-R-lem}, the similar statement holds in a neighborhood of a curve with a single Ramond node.
This implies that $\pi_*(\AA_X/\AA_{X/\ov{\SS}})$ is locally free over $\SS'\sub \ov{\SS}$, and the divisor of the Berezinian of \eqref{boundary-eq} over $\SS'$ is
$\De\cap \SS'$.

Nex, let us study the situation near a point $[X_0]$ of $\ov{\SS}$, such that $[X_0]$ has several nodes $q_1,\ldots,q_k$.
Using the fact that the map from deformations of $[X_0]$ to the product of deformation spaces of the nodes is smooth, we see that there exists
an \'etale neighborhood $B$ of $[X_0]$, together with a smooth map $(t_1,\ldots,t_k):B\to \A^k$ such that the function $t_i$ corresponds the induced deformation of
the node $q_i$.

Let us consider the normal crossing divisor $D=(t_1\ldots t_k=0)$ in $B$, and let $\TT_{B,D}\sub \TT_B$ be the subsheaf of derivations preserving $D$.
Note that $\TT_{B,D}=\cap \TT_{B,D_i}$ where $D_i$ is the divisor $t_i=0$.

Let us consider the restriction $\pi:X\to B$ of the universal family and set
$$\ov{\AA}_X:=\AA_X/\AA_{X/B}\sub \pi^*\TT_B.$$
Let $U_i$ be an open neighborhood of $q_i$ in $X$, and let $\pi_i=\pi|_{U_i}$. Then, as we have seen above,
we have
$$\ov{\AA}_X|_{U_i}=\pi_i^{-1}\TT_{B,D_i}.$$
Thus, for each $i=1,\ldots,k$, we have an inclusion
$$\pi_*(\ov{\AA}_X)\sub \pi_{i*}(\ov{\AA}_X|_{U_i})=\TT_{B,D_i}\sub \TT_B.$$
Hence, we deduce the inclusion
$$\pi_*(\ov{\AA}_X)\sub \TT_{B,D}.$$

On the other hand, we claim that there is an inclusion
$$\pi^{-1}\TT_{B,D}\sub \ov{\AA}_X.$$
Indeed, it is enough to check this over each $U_i$. But then we have
$$\pi^{-1}\TT_{B,D}|_{U_i}\sub \pi_i^{-1}\TT_{B,D_i}=\ov{\AA}_X,$$
as required.
Hence, passing to $\pi_*(?)$, we derive the inclusion
$$\TT_{B,D}=\pi_*\pi^{-1}\TT_{B,D}\sub \pi_*\ov{\AA}_X.$$
as claimed.

Thus, we get $\pi_*\ov{\AA}_X=\TT_{B,D}$.
Hence, $\pi_*\ov{\AA}_X$ is locally free and the Berezinian of the morphism $\TT_{B,D}\to \TT_B$ has the required form.
\end{proof}

\subsection{NS and Ramond boundary components as effective Cartier divisors}\label{De-NS-R-sec}

Note that $\De\cap \SS'$ is the disjoint union of two Cartier divisors, the one supported on stable supercurves with one NS node and the one supported on
those with a Ramond node. We want to extend these two divisors on $\SS'$ to effective Cartier divisors $\De_{NS}$ and $\De_R$ on $\ov{\SS}$ such that
$\De=\De_{NS}+\De_R$.

First, we are going to define the divisor $\De_{NS}$ giving the NS component.
For this let us consider the structure map
$$\de:\Om_{X/\ov{\SS}}\to \om_{X/\ov{\SS}}(R).$$
As we have seen before, it is surjective away from the nodes.
It is also surjective on Ramond nodes.
Let us consider the ideal sheaf on $\ov{\SS}$,
$$\II_\de:=\Ann \pi_*\coker(\de),$$
supported on the locus of stable supercurves with at least one NS node.

\begin{prop}

(i) The ideal sheaf $\II_\de$ defines an effective Cartier divisor $\De_{NS}$ on $\ov{\SS}$, which coincides with $\De$
in a neighborhood of any point $[X_0]$ corresponding to a stable supercurve $X_0$ with only NS nodes.

\noindent
(ii) There exists a unique effective Cartier divisor $\De_R$ supported on the locus of stable supercurves with at least one Ramond node,
such that
$$\De=\De_{NS}+\De_R.$$
If $X_0$ is a stable supercurve with NS nodes $q_1,\ldots,q_r$ and Ramond nodes $q_{r+1},\ldots,q_k$, then there exists an \'etale neighborhood $B$ of
$[X_0]$ in $\ov{\SS}$ with a smooth morphism $t_1,\ldots,t_k:B\to \A^k$ such that $t_1\ldots t_r$ is an equation of $\De_{NS}$ and $t_{r+1}\ldots t_k$ is an equation
of $\De_R$.

\end{prop}

\begin{proof} Let $X_0$ be a stable supercurve with NS nodes $q_1,\ldots,q_r$ and Ramond nodes $q_{r+1},\ldots,q_k$.
Consider an \'etale neighborhood $B$ of $[X_0]$, equipped with a smooth map $(t_1,\ldots,t_k):B\to \A^k$,
such that $t_i$ gives the universal deformation of the node $q_i$.

Then the ideal of $\De$ is generated by $t_1\ldots t_k$.

On the other hand, using the description of $\om_{X/S}$ by generators and relations (see Lemma \ref{NS-om-lem}), we see that in the formal neighborhood of each NS node $q_i$,
$\coker(\de)$ is generated by $s_0$ subject to the relations
$$\th_1s_0=\th_2s_0=z_1s_0=z_2s_0=t_is_0=0.$$
It is easy to deduce from this that the annihilator of $\pi_*\coker(\de)$ is generated by $t_1\ldots t_r$.
This implies all the assertions.

\end{proof}

\subsection{The NS node boundary components}\label{NS-boundary-gluing-sec}

We will use the following construction of {\it gluing two superschemes along a closed subscheme}.

Assume that $X$, $X'$ and $Y$ are superschemes, $i:Y\to X$ and $i:Y\to X'$ are closed embeddings.
We want to define a new superscheme $Z$ by gluing $X$ and $X'$ along $Y$. As a topological space we
can define $Z$ to the usual gluing of the topological spaces of $X$ and $X'$ along $Y$, so that we have
closed embeddings $j:X\to Z$, $j':X'\to Z$, so that $k:=j_1\circ i=j_2\circ i$. Hence, we have two homomorphisms
of sheaves of rings
$$\phi:j_*\OO_X\to k_*\OO_Y, \ \ \phi':j'_*\OO_{X'}\to k_*\OO_Y,$$
and we define $\OO_X$ to be the subsheaf of $j_*\OO_X\oplus j'_*\OO_{X'}$, namely, the preimage of the diagonal
$k_*\OO_Y\sub k_*\OO_Y\oplus k_*\OO_Y$ under $\phi\oplus \phi'$.

We can apply this to gluing two stable supercurves along NS-punctures.

Suppose $X/S$ and $X'/S$ is a pair of stable supercurves with NS-punctures $P\sub X$ and $P'\sub X'$.
We have an isomorphism $P\simeq S\simeq P'$, so we can define a new superscheme $Z/S$ by gluing $X$ and $X'$
along $P\simeq P'$.

\begin{lemma}
The glued superscheme $Z/S$ has a natural stable supercurve structure such that the derivation $\de$ on $Z$ is defined as the composition
$$\OO_Z\to j_*\OO_X\oplus j'_*\OO_{X'}\rTo{(j_*\de,j'_*\de')} j_*\om_{X/S}\oplus j'_*\om_{X'/S}\to \om_{Z/S}.$$
\end{lemma}

\begin{proof}
If $U\sub X$, $U'\sub X'$ are smooth loci, then $(U\setminus P)\sqcup (U'\setminus P')$ is an open subset of $Z$, which is a smooth supercurve over $S$.
Next, we need to check that in the case when $S$ is a point, $\de^-$ induces an isomorphism of $\OO_Z^-$ with $\om_Z^-$.
Since the base is even, we have identifications
$$\OO_X=\OO_C\oplus L, \ \ \OO_{X'}=\OO_{C'}\oplus L',$$
and the smooth marked points $P\sub C$, $P'\sub C'$, so that the embedding of $P$ into $X$ corresponds to the projection
$$\OO_X\to \OO_C\to \OO_P.$$
Note that $L$ and $L'$ are generalized spin-structures, i.e., we have an isomorphism $L\rTo{\sim}\und{\Hom}(L,\om_C)$ (which corresponds to $\de^-$ on $X$),
and similarly, for $L'$. Furthermore, we know that $L$ (resp., $L'$) is locally free near $P$ (resp., $P'$).

The glued superscheme $Z$ has the underlying nodal curve $Z_0$, which is glued from $C$ and $C'$ along $P\simeq P'$,
and
$$\OO_Z\simeq \OO_{Z_0}\oplus j_*L\oplus j'_*L'.$$
It is well known that the natural map
$$j_*L\oplus j'_*L'\to \und{\Hom}(j_*L\oplus j'_*L',\om_{Z_0})$$
is an isomorphism, i.e., $j_*L\oplus j'_*L'$ is a generalized spin-structure.
It is easy to see that above map is precisely $\de^-$, so this proves that $Z/S$ is a stable supercurve.
\end{proof}

Let $\ov{\SS}_{g;m,n}$ denote the moduli superspace of stable supercurves of genus $g$ with $m$ NS marked points and $n$ Ramond marked points.
Then we can apply the above gluing construction to the pair of families of stable supercurves over
$\ov{\SS}_{g_1;m_1+1,n_1}\times \ov{\SS}_{g_2;m_2+1,n_2}$, pulled back from each factor and using the last NS puncture on each of them.
This leads to a morphism
\begin{equation}\label{NS-sep-node-morphism}
\ov{\SS}_{g_1;m_1+1,n_1}\times \ov{\SS}_{g_2;m_2+1,n_2}\to \ov{\SS}_{g_1+g_2;m_1+m_2,n_1+n_2}.
\end{equation}

Similarly, if $X/S$ is a stable supercurve with two disjoint NS-punctures $P,P'\sub X$, then we can glue $P$ with $P'$ and get a new stable supercurve with
a non-separating node. This leads to a morphism
\begin{equation}\label{NS-nonsep-node-morphism}
\ov{\SS}_{g;m+2,n}\to \ov{\SS}_{g+1;m,n}
\end{equation}

\begin{lemma}\label{NS-boundary-tangent-emb-lem}

Both morphisms \eqref{NS-sep-node-morphism} and \eqref{NS-nonsep-node-morphism} induce embeddings of codimension $1|0$ on tangent spaces and factor
through the divisor $\De_{NS}$. Furthermore, near a stable supercurve with a single separating (resp., non-separating)
NS node, the divisor $\De_{NS}$ coincides with the schematic image of \eqref{NS-sep-node-morphism} (resp., \eqref{NS-nonsep-node-morphism}).

\end{lemma}

\begin{proof}

The idea is to use exact sequences \eqref{def-curve-nodes-ex-seq}.

Let $X$ be a stable supercurve with punctures and a fixed NS-node $q\in X$, and let $\rho:\wt{X}\to X$ be the normalization at $q$, equipped with the two NS punctures over $q$.
Then by Proposition \ref{no-inf-aut-prop}(i), we have
$$\AA_X\simeq \rho_*\AA_{\wt{X}}$$
(where we take into account all the punctures on both sides).
Hence, the natural map
$$H^1(\wt{X},\AA_{\wt{X}})\to H^1(X,\AA_X)$$
is an isomorphism.

Let $q_1,\ldots,q_m$ be the nodes of $X$ different from $q$. Then we have a morphism of exact sequences
\begin{diagram}
0&\rTo{}&H^1(\wt{X},\AA_{\wt{X}})&\rTo{}& T_{\Def(\wt{X})} &\rTo{}& \bigoplus_{i=1}^m T_{\Def(\OO_{X,q_i})} &\rTo{}& 0\\
&&\dTo{}&&\dTo{}&&\dTo{}\\
0&\rTo{}&H^1(X,\AA_X)&\rTo{}& T_{\Def(X)} &\rTo{}& T_{\Def(\OO_{X,q})}\oplus\bigoplus_{i=1}^m T_{\Def(\OO_{X,q_i})} &\rTo{}& 0
\end{diagram}
in which the right vertical arrow is the natural inclusion of codimension $1|0$. Since the left vertical arrow is an isomorphism, the assertion about the
map of tangent spaces follows.

To check that our morphisms factor through $\De_{NS}$, we have to check that the pull-back of the morphism $\OO\to \OO(\De_{NS})$ is zero.
Since the sources are smooth, it is enough to check this generically, so we can consider a neighborhood of the point corresponding to a supercurve with a single
NS node. Then we know that the local equation of $\De_{NS}$ will be $(t=0)$, where $t$ corresponds to the map to the universal deformation of the node.
Since the deformation of the node given by the source of the maps \eqref{NS-sep-node-morphism} and \eqref{NS-nonsep-node-morphism} is trivial, this proves that the pull-back of $t$ with respect to one of these maps is zero. Using the fact that the morphism on tangent spaces is an embedding with the image which is the orthogonal to $dt$, we deduce
that near this point $\De_{NS}$ is the schematic image of the morphism.

\end{proof}

\subsection{The Ramond node boundary components}\label{R-boundary-gluing-sec}

Gluing along two Ramond punctures is more subtle.

First, we point out a certain ``residual structure" we have on each Ramond puncture.
Let $R\sub X$ be a Ramond puncture in a stable supercurve $X/S$.

\begin{lemma}\label{R-puncture-structure-lem}
The map
$$\de|_R:\Om_{X/S}|_R\to \om_{X/R}(R)|_R\simeq \om_{R/S}$$
factors through the canonical projection $\Om_{X/S}|_R\to \Om_{R/S}$, and induces
an isomorphism
$$\ov{\de}_R:\Om_{R/S}\to \om_{R/S}.$$
This induces a trivialization of $(\Om_{R/S})^{\ot 2}$ (or equivalently, of $\om_{R/S}^{\ot 2}$)
which is locally given by $(d\th)^2$, for $\th$ such that
$\OO_R=\OO_S[\th]$,
\end{lemma}

\begin{proof}
Recall that in appropriate \'etale local coordinates $(z,\th)$, $R$ is given by $(z=0)$ and we have
$$\de(dz)=z\th [\frac{dz}{z}|d\th], \ \ \de(d\th)=  [\frac{dz}{z}|d\th].$$
It follows that $\de|_R(dz)=0$, and
the induced map $\ov{\de}_R$ is given by
$$\ov{\de}_R(d\th)=\ov{b},$$
where $\ov{b}$ is the generator of $\om_{R/S}$ corresponding to $[\frac{dz}{z}|d\th]$.

Since the relative dimension of $R$ is $0|1$, we have in fact an isomorphism
$\om_{R/S}^{-1}\simeq \Om_{R/S}$. Thus, we can view $\ov{\de}_R$ as a trivializing section of $(\Om_{R/S})^{\ot 2}$.

\end{proof}

The above Lemma implies that every Ramond puncture $R$ has a preferred system of \'etale local relative coordinates $\th_i$, such that over intersections one has
$\th_j=\pm \th_i+a_{ij}$, where $a_{ij}$ are functions on the base. Namely, we require that $(d\th_i)^2$ is the canonical trivialization of $(\Om_{R/S})^{\ot 2}$.
In other words, we have a canonical principal bundle $\PP_R\to S$ with the structure supergroup $\Z/2\ltimes \A^{0|1}$.

We can restate the above structure in more invariant terms. Let $\pi:R\to S$ denote the projection.
First, we observe that
\begin{equation}\label{PhiR-def}
\Phi_R:=\pi_*\OO_R/\OO_S\simeq \Ber^{-1}\pi_*\OO_R
\end{equation}
is a line bundle of rank $0|1$ on $S$. Furthermore, we have a canonical isomorphism of odd line bundles on $R$,
$$\pi^*\Phi_R\rTo{\sim} \Om_R$$
induced by the de Rham differential $\OO_S/\OO_R\to \Om_R$. Furthermore, the trivialization of $(\Om_R)^{\ot 2}$
in Lemma \ref{R-puncture-structure-lem} actually comes from the canonical trivialization of
$\Phi_R^2$ on $S$.

Let $\Theta\sub \tot(\Phi_R)$ be the $\Z/2$-torsor over $S$ corresponding to $\Phi_R$.
Then the principal bundle $\PP_R\to S$ can be identified with the preimage of $\Theta$ in $\tot(\pi_*\OO_R)$ under
the natural projection $p:\pi_*\OO_R\to \pi_*\OO_R/\OO_S$.

\begin{lemma}\label{Aut-PhiR-lem}
The group scheme $\Aut(\Phi_R)\to S$ can be identified with the group of automorphisms of $\pi_*\OO_R$, which are
identity on $\OO_S$, and induce $\pm \id$ on $\pi_*\OO_R/\OO_S$. This group scheme is an extension of $\Z/2$ by
the line bundle $\Phi_R^{-1}$.
\end{lemma}

Now suppose $X/S$ is a (possibly disconnected) stable supercurve  with two R-punctures $R\sub X$ and $R'\sub X$,
equipped with an isomorphism $R\simeq R'$ over $S$.
Then we can glue $X$ with itself along $R\simeq R'$ into a superscheme $Z$ over $S$ equipped with a finite morphism $j:X\to Z$.

\begin{lemma}\label{Ramond-gluing-lem}
Assume that the isomorphism $\a:R\rTo{\sim} R'$ is such that
the induced isomorphism
$$\a^*\om_{R'/S}^{\ot 2}\rTo{\sim} \om_{R/S}^{\ot 2}$$
is equal to $-1$, where we use the trivializations of $\om_{R/S}^{\ot 2}$ and
$\om_{R'/S}^{\ot 2}$ coming from the supercurve structure on $X$.
Then the glued superscheme $Z/S$ has a natural stable supercurve structure, such that the derivation $\de$ is
uniquely determined from the commutative diagram
\begin{diagram}
\OO_Z &\rTo{\de}& \om_{Z/S}\\
\dTo{r}&&\dTo{}\\
j_*\OO_X&\rTo{j_*\de}& j_*\om_{X/S}(R)
\end{diagram}
\end{lemma}

\begin{proof} To check that $\de$ is well defined, we need to check that the image of the composition
$j_*\de\circ r$ belongs to $\om_{Z/S}$.
For this we can argue locally. Let $(z,\th)$ be the standard local coordinates on $X$ near the puncture $R$,
so that $R$ is given by $(z=0)$, and
$$\de(dz)=z\th [\frac{dz}{z}|d\th], \ \ \de(d\th)=  [\frac{dz}{z}|d\th].$$
Let also $(z',\th')$ be similar coordinates near $R'$. Then $\a^*(d\th')^2=-d\th$, so changing $\th$ to $\pm\th+a$, we can assume
that $i\a^*(\th')=\th$.
Let us set $\ov{\th}=i\th'$. Then we have
$$\de(dz')=z'\ov{\th} [\frac{dz}{z}|d\ov{\th}], \ \ \de(d\ov{\th})=-[\frac{dz'}{z'}|d\ov{\th}].$$

Then we have relative coordinates on the glued scheme $Z$, $(z,z',\th)$, where $zz'=0$, $\th$ restricts to $\th$ near $R$ and to $\ov{\th}$ near $R'$.
Since $\de(z)$ has no pole at $R$, we see that $(\de(z),0)$ belongs to $\om_{Z/S}$. Similarly, $(0,\de'(z'))$ belongs to $\om_{Z/S}$.
Finally,
$$(\de(\th),\de'(\ov{\th}))=([\frac{dz}{z}|d\th],-[\frac{dz'}{z'}|d\ov{\th}]),$$
which comes from a section of $\om_{Z/S}$.

Over the point, our construction can be recast as follows. We start with a Ramond spin curve $(C,p,p',L)$, together with an identification
$$\a:L|_p\simeq L|_{p'},$$
such that $\a^2=-1$ (where we use trivializations of $\om_C(p)|_p$ and $\om_{C}(p')|_{p'}$).
We glue $p$ with $p'$ and get a nodal curve $Z_0$, and then descend $L$ to a line bundle $\LL$ over $Z_0$ using $\a$.
Then we have a natural isomorphism $\LL^{\ot 2}\simeq\om_{Z_0}$, and one can easily check that the corresponding isomorphism
$\LL\rTo{\sim} \und{\Hom}(\LL,\om_{Z_0})$ is induced by $\de^-$, where $\de$ is defined as above.
\end{proof}

The choice of an isomorphism $\a$ can be interpreted in terms of the principal $\Z/2\ltimes\A^{0|1}$-bundles $\PP_R$ and $\PP_{R'}$ as follows.
For every $c\in \C^*$ let us denote by $[c]$ the automorphism of $\Z/2\ltimes\A^{0|1}$ given by the rescaling by $c$ on $\A^{0|1}$ (and trivial on $\Z/2$).
Then a choice of $\a$ is equivalent to a choice of an isomorphism of $\Z/2\ltimes\A^{0|1}$-bundles,
$$\PP_R\to [i]_*\PP_{R'},$$
where $[i]_*\PP_{R'}$ is the push-out of $\PP_{R'}$ with respect to the automorphism $[i]$.

We can apply the above construction to the two last Ramond punctures $R_{n+1}$ and $R_{n+2}$ on the universal stable supercurve over
$\ov{\SS}_{g;m,n+2}$.
Let $\PP_{n+1}\to \ov{\SS}_{g;m,n+2}$ and $\PP_{n+2}\to \ov{\SS}_{g;m,n+2}$ be the corresponding
principal $\Z/2\ltimes \A^{0|1}$-bundles.
Let $\Isom(\PP_{n+1},[i]_*\PP_{n+2})\to \ov{\SS}_{g;m,n+2}$ denote the bundle of isomorphisms between these $\Z/2\ltimes \A^{0|1}$-bundles.
Note that it is also a principal bundle with the group $\Z/2\times \A^{0|1}$.

Then the above gluing construction gives a family of stable supercurves with a Ramond node over $\Isom(\PP_{n+1},[i]_*\PP_{n+2})$, so we get a morphism
\begin{equation}\label{Ramond-nonsep-node-morphism}
\Isom(\PP_{n+1},[i]_*\PP_{n+2})\to \ov{\SS}_{g+1;m,n}
\end{equation}

Similarly, we can glue the last Ramond punctures $R,R'$ on the stable supercurves $X,X'$ over the moduli space
$\ov{\SS}_{g_1;m_1,n_1+1}\times \ov{\SS}_{g_2;m_2,n_2+1}$ pulled back from each factor.
This gives a morphism
\begin{equation}\label{Ramond-sep-node-morphism}
\Isom(\PP_R,[i]_*\PP_{R'})\to \ov{\SS}_{g_1+g_2;m_1+m_2,n_1+n_2}.
\end{equation}
Note that in this case $n_1$ and $n_2$ are odd, so $n_1+n_2\ge 2$.

\begin{lemma}
Both morphisms \eqref{Ramond-nonsep-node-morphism} and \eqref{Ramond-sep-node-morphism} induce embeddings of codimension $1|0$ on tangent spaces
and factor through the divisor $\De_R$. Furthermore, near a stable supercurve with a single non-separating Ramond node the divisor $\De_R$ coincides with the schematic image
of \eqref{Ramond-nonsep-node-morphism}.
\end{lemma}

\begin{proof}
Let $X$ be the stable supercurve obtained by gluing two Ramond punctures $R, R'$ on a stable supercurve $\wt{X}$, via an isomorphism
$\a:R\to R'$ as in Lemma \ref{Ramond-gluing-lem}.
Note that we have an exact sequence
$$0\to \Pi\C\to T_{\Def(\wt{X},\a)}\to T_{\Def(\wt{X})}\to 0,$$
where the odd line $\Pi\C$ corresponds to infinitesimal deformations of $\a$.
By Proposition \ref{no-inf-aut-prop}(i), we have
$$\AA_X^+\simeq \rho_*\AA_{\wt{X}}^+.$$
As in Lemma \ref{NS-boundary-tangent-emb-lem} we deduce from this that
$$T^+_{\Def(\wt{X},\a)}\to T^+_{\Def(X)}$$
is an embedding of codimension $1$.

Next, we claim that the map between the spaces of odd infinitesimal deformations,
$$T^-_{\Def(\wt{X},\a)}\to T^-_{\Def(X)}$$
is an isomorphism. Indeed, we know that every odd deformation of $X$ is locally trivial, i.e.,
$$T^-_{\Def(X)}\simeq H^1(X,\AA_X^-).$$
Also, by Proposition \ref{no-inf-aut-prop}(i),
we have an exact sequence
$$0\to \C\to H^1(X,\AA_X^-)\to H^1(\wt{X},\AA_{\wt{X}}^-)\to 0$$
so
$$\dim T^-_{\Def(X)}= \dim H^1(\wt{X},\AA_{\wt{X}}^-)+1=\dim T^-_{\Def(\wt{X},\a)}.$$

It remains to observe that every locally trivial deformation of $X$
can be lifted to a deformation of $(\wt{X},\a)$.
\end{proof}

\section{Canonical line bundle on the moduli space of stable supercurves}\label{canonical-bundle-sec}

In this section we will calculate the canonical line bundle of $\ov{\SS}_{g,n_{NS},n_R}$, eventually proving Theorem B.

\subsection{What Kodaira-Spencer isomorphism gives for the canonical bundle}

Set $\ov{\SS}=\ov{\SS}_{g,n_{NS},n_R}$. We are interested in the canonical line bundle $K_{\ov{\SS}}=\Ber^{-1}(\TT_{\ov{\SS}})$.
Over the smooth locus $\SS\sub \ov{\SS}$, from the Kodaira-Spencer isomorphism we get an isomorphism (see Sec.\ \ref{Kodaira-punct-sec})
\begin{equation}\label{smooth-can-KS-formula-eq}
\ber(KS_{\ov{\SS}}):K_{\SS}\rTo{\sim} \Ber^{-1}(R^1\pi_*\LL(X,P_\bullet,R_\bullet))\simeq \Ber(R\pi_*\LL(X,P_\bullet,R_\bullet)),
\end{equation}
where
$$\LL(X,P_\bullet,R_\bullet):=\om_{X/S}^{-2}(-\sum_{i\in I} D_i-2\sum_{j\in J} R_j).$$
Recall that
here $\om_{X/S}^{-2}$ is the line bundle defined in Theorem \ref{can-square-thm}, and
$D_i$ are the divisors on the universal curve $X$ associated with the NS punctures.

Our study in Sec.\ \ref{KS-map-sec}
of the behavior of the Kodaira-Spencer map in degenerating families of supercurves leads to the following identification of $K_{\ov{\SS}}$.

\begin{prop}\label{stable-can-KS-formula-prop}
One has a natural isomorphism
$$\ber(KS_{\ov{\SS}})^{-1}:K_{\ov{\SS}}\rTo{\sim} \Ber(R\pi_*\LL(X,P_\bullet,R_\bullet))(-\De),$$
where $\De$ is the boundary divisor.
\end{prop}

\begin{proof}
We just have to check that isomorphism \eqref{smooth-can-KS-formula-eq} acquires simple poles at all generic points of $\De$.
But this follows from the results of Sec. \ \ref{KS-map-sec} and our local description of $\De$ at points corresponding to stable
supercurves with single nodes.
\end{proof}

Next, we will study the line bundle $\Ber(R\pi_*\LL(X,P_\bullet,R_\bullet))$. We begin with the case
when there are no punctures. Then the above identification of the canonical bundle becomes

$$\ber(KS_{\ov{\SS}})^{-1}:K_{\ov{\SS}}\rTo{\sim} \Ber(R\pi_*(\om_{X/S}^{-2})).$$

\subsection{Isomorphisms for Berezinian bundles}\label{super-Deligne-sec}

Let $\pi:X\to S$ be a smooth proper morphism of relative dimension $1|1$. Then we can define Deligne's symbol
$\lan L_1$, $L_2\ran$ of a pair of line bundles of rank $1|0$
over $X$ similarly to the even case, so that for a relative effective Cartier divisor $D$
one has
$$\lan \OO_X(D),L\ran\simeq \Ber(\pi_*(L|_D))/\Ber(\pi_*\OO_D).$$

If $\BB(L)$ denotes the Berezinian of the derived push-forward of $L$ then one has, as in the classical case,
$$\BB(L_1\ot L_2)\ot\BB(\OO)\simeq \BB(L_1)\ot\BB(L_2)\ot\lan L_1,L_2\ran.$$
However, in the supercase,
for any line bundles $L$ and $L'$ of rank $1|0$,
there is a canonical isomorphism
\begin{equation}\label{Ber-L-D-eq}
\a_D:\Ber(\pi_*L|_D)\rTo{\sim}\Ber(\pi_*(L'|_D)),
\end{equation}
induced by any local isomorphism $L\to L'$ (the point is that the Berezinian of a scalar automorphism of a linear space of rank $1|1$ is trivial). Hence, the Deligne's symbol is canonically trivial, so we get
\begin{equation}\label{super-Deligne-eq}
\BB(L_1\ot L_2)\ot\BB(\OO)\simeq \BB(L_1)\ot\BB(L_2)
\end{equation}
(see \cite{Voronov}).

More explicitly, locally over $S$ we can pick a relative positive divisor $D$ such that $L_1L_2(D)$, $L_2(D)$ and $\OO_X(D)$ have no $R^1\pi_*$,
and assume also that we have an even section $s\in H^0(X,L_2(D))$, fiberwise regular, vanishing on a relative divisor $E$.
Then we have the resolutions
\begin{align*}
&R\pi_*(\OO_X): & [\pi_*L_2(D)\to \pi_*L_2(D)|_E], \\
&R\pi_*(L_1): & [\pi_*L_1L_2(D)\to \pi_*L_1L_2(D)|_E], \\
&R\pi_*(L_2): & [\pi_*L_2(D)\to \pi_*L_2(D)|_D], \\
&R\pi_*(L_1L_2): & [\pi_*L_1L_2(D)\to \pi_*L_1L_2(D)|_D].
\end{align*}
Using these resolutions we get an isomorphism
\begin{align*}
&\BB(L_1L_2)\ot \BB(\OO_X)=\BB(L_1L_2(D))\ot \BB(L_2(D))\ot \Ber^{-1}(\pi_*L_1L_2(D)|_D)\ot \Ber^{-1}(\pi_*L_2(D)|_E)\\
&\rTo{\id\ot\id\ot\a^{-1}_D\ot\a^{-1}_E}
\BB(L_1L_2(D))\ot \BB(L_2(D))\ot \Ber^{-1}(\pi_*L_2(D)|_D)\ot \Ber^{-1}(\pi_*L_1L_2(D)|_E)=\BB(L_1)\ot \BB(L_2).
\end{align*}

Note that isomorphism \eqref{super-Deligne-eq} can be used to calculate $\BB(L_1L_2)$ when $L_1$ and $L_2$ are not necessarily of
rank $1|0$: for a line bundle $L$ of rank $0|1$ we use the isomorphism
$$\BB(\Pi L)\simeq \BB(L)^{-1}.$$

For example, for a line bundle $L$ of rank $1|0$ one has
$$\BB(L^3)\simeq \BB(L)^3/\BB(\OO)^2.$$
Applying this to $L=\Pi \om_X$, where $X/S$ is a smooth supercurve,
one gets the analog of Mumford isomorphism in the supercase (see \cite{Voronov}),
\begin{equation}\label{super-Mumford-eq}
\BB(\om_{X/S}^3)\simeq \BB(\om_{X/S})^3\ot \BB(\OO_X)^2\simeq \BB(\om_{X/S})^5,
\end{equation}
since $\BB(L)\simeq \BB(\om_{X/S}\ot L^{-1})$ by Grothendieck-Serre duality.
Thus, denoting $\Ber_i:=\BB(\om_{X/S}^i)$, we can write this isomorphism as
$$\Ber_3\simeq \Ber_1^5.$$

We will need an extension of isomorphism \eqref{super-Deligne-eq} to families of stable supercurves $\pi:X\to S$ and to the case when $L_1$ is replaced by a not necessarily
locally free sheaf. For a coherent sheaf $\FF$ over $X$, flat over $S$, we denote by $\BB(\FF)$ the Berezinian of the perfect complex $R\pi_*(\FF)$.

\begin{lemma}\label{super-Deligne-nonfree-lem}
Let $\pi:X\to S$ be a stable supercurve, $\FF$ a coherent sheaf on $X$ flat over $S$.
Assume that $\FF$ is locally free of rank $1|0$ over the smooth locus of $\pi$. Then for any line bundle $L$ of rank $1|0$ on $X$, one has
a canonical isomorphism
\begin{equation}\label{super-Deligne-sheaf-eq}
\BB(\FF\ot L)\simeq \BB(\FF)\ot \BB(L)\ot \BB(\OO_X)^{-1}.
\end{equation}
\end{lemma}

\begin{proof} We can use the similar recipe as above to construct this isomorphism locally over $S$: we simply replace $L_1$ by $\FF$, $L_2$ by $L$ and
take $D$ with support in the smooth locus of $\pi$, and a section $s\in H^0(X,L(D))$ with the vanishing divisor $E$, also contained in the smooth locus of $\pi$.
\end{proof}

\subsection{Behavior of the super Mumford isomorphism near the NS boundary divisor}\label{super-Mum-NS-sec}

Now we want to look at the behavior of the super Mumford isomorphism \eqref{super-Mumford-eq}
near the generic point of boundary divisors. In the case of a Ramond node it extends to an isomorphism, since $\om_{X/S}$ is
still a line bundle. Thus, it remains to study the boundary component where one NS node appears. So let
$\pi:X\to S$ be a family of stable supercurves as in Sec.\ \ref{super-KS-NS-sec}, so $S$ is a formal super-polydisk with a
special coordinate $t$, $S_0\sub S$ is the divisor $t=0$, $q:S_0\to X_0\sub X$ is the relative node over $S_0$.

Let $S'=S\setminus S_0$, and let $X'=\pi^{-1}(S')$ be the corresponding family of smooth supercurves.
First, we need to explain how to extend the line bundles $\BB(\om_{X'/S'}^3)$ and $\BB(\OO_{X'})$ to $S$.
For the second one this is straightforward: the extension is given by $\BB(\OO_X)$.

For any integer $n$ let us set
$$\om_{X/S}^n:=j_*\om_{U/S}^n,$$
where $U\sub X$ is the smooth locus of $\pi$.
Recall that $\om_{X/S}^2$ is a line bundle on $X$ (see Theorem \ref{can-square-thm}). Thus, we have
$$\om_{X/S}^3=j_*\om_{U/S}^3\simeq \om_{X/S}^2\ot j_*\om_{U/S}\simeq \om_{X/S}^2\ot \om_{X/S}.$$
This is a coherent sheaf, flat over $S$, so we can take $\BB(\om_{X/S}^3)$ as the desired extension of
$\BB(\om_{X'/S'}^3)$.

Note that by Grothendieck-Serre duality, we have an isomorphism of line bundes on $S$,
$$\BB(\om_{X/S})\simeq \BB(\OO_X).$$

Since $\om_{X/S}$ is a coherent sheaf flat over $S$, we deduce from Lemma \ref{super-Deligne-nonfree-lem} an isomorphism
$$\BB(\Pi\om_{X/S}^3)\simeq \BB(\Pi\om_{X/S})\ot \BB(\om_{X/S}^2)\ot \BB(\OO_X)^{-1}\simeq \BB(\om_{X/S}^2)\ot \BB(\OO_X)^{-2},$$
or equivalently,
\begin{equation}\label{BB-om3-X-Mum-eq}
\BB(\om_{X/S}^3)\simeq \BB(\om_{X/S}^2)^{-1}\ot \BB(\OO_X)^2.
\end{equation}

Thus, our problem reduces to studying the behavior of the isomorphism on $X'$,
$$\BB(\om_{X'/S'}^2)\simeq \BB(\Pi\om_{X'/S'})^2\ot \BB(\OO_{X'})^{-1}\simeq \BB(\OO_{X'})^{-3}$$
near the divisor $S_0\sub S$. To this end we look more carefully at the recipe for this isomorphism outlined in Sec.\ \ref{super-Deligne-sec}.

We can pick a sufficiently positive divisor $D$ in the smooth locus of $\pi$ and also an even section $s$ of $\Pi\om_{X/S}(D)$
with the zero locus $E$. Furthermore, we want to make a special choice of $s$.

\begin{lemma}\label{choice-of-section-lem}
There exists an exact sequence of $\OO_X$-modules,
$$0\to \om^{\reg}_{X/S}\to \om_{X/S}\to q_*\OO_{S_0}\to 0$$
such that in the formal neighborhood of $q(S_0)$, $\om^{\reg}_{X/S}$ is the $\OO_X$-submodule generated by the sections $s_1$ and $s_2$
given by \eqref{s1-s2-s0-generators}.
Furthermore, we can choose an even section $s$ of $\Pi\om^{\reg}_{X/S}(D)$ in such a way that in the formal neighborhood of $q$ one has
$$s=v\cdot (s_1+u\cdot s_2),$$
for some invertible functions $u$ and $v$ that are congruent to $1$ modulo the ideal of $q(S_0)$.
By making a change of variables
$$z_1\mapsto u^{-1}z_1,  \ \ z_2\mapsto uz_2, \ \ \th_1\mapsto \th_1, \ \ \th_2\mapsto u\th_2$$
we can achieve that $u=1$.
\end{lemma}

\begin{proof}
In the formal neighborhood of $q$, the quotient of $\om_{X/S}/(\OO_X s_1+\OO_X s_2)$ is generated by $s_0$ and is isomorphic to $q_*\OO_{S_0}$.
This implies the first assertion. Furthermore, using Lemma \ref{NS-om-lem}, we see that
$\om^{\reg}_{X/S}\ot q_*\OO_{S_0}$ is a free $q_*\OO_{S_0}$-module generated by the images of $s_1$ and $s_2$.
For sufficiently positive $D$, the map
$$H^0(X,\om^{\reg}_{X/S}(D))\to H^0(X,\om^{\reg}_{X/S}\ot q_*\OO_{S_0})$$
is surjective, so we can choose an odd section $s\in H^0(X,\om^{\reg}_{X/S}(D))$ such that
$$s\equiv s_1+s_2\mod J_{q(S_0)}.$$
Thus, in the formal neighborhood of $q$, we can write $s=u_1s_1+u_2s_2$ where $u_i\equiv 1\mod J_{q(S_0)}$.

It is easy to check that the given change of variables transforms $s$ in the desired way.
\end{proof}

Now we define coherent sheaves $\FF$ and $\GG$ on $X$ by the exact sequences
$$0\to \OO_X\rTo{s}\Pi \om_{X/S}(D)\to \FF\to 0,$$
$$0\to \Pi \om_{X/S} \rTo{\cdot s} \om_{X/S}^2(D)\to \GG\to 0.$$
Let $\hat{\FF}$ and $\hat{\GG}$ denote
the completions of $\FF$ and $\GG$ at $q$

\begin{lemma}\label{F-G-completions-lem}
(a) The space $\hat{\FF}$ is a free $\OO_S$-module with the basis $s_1, s_0$.

\noindent
(b) We have an isomorphism $\hat{\GG}\simeq \OO_Z$, where $Z\sub X$ is a sub-superscheme given by
$$z_1=-t, \ \ z_2=t, \ \ \th_1+\th_2=0.$$
In particular, $\hat{\GG}$ is a free $\OO_S$-module with the basis $e, \th_1e$,
where $e$ is the local generator of $\om_{X/S}^2$ given by \eqref{e-sec-eq}.
The sheaf $\hat{\FF}$ is also scheme-theoretically supported on $Z$.

\noindent
(c) The Berezinian of the map of $\OO_S$-modules
$$s_1\cdot ?: \hat{\FF}\to \hat{\GG}$$
is equal to $ft$, where $f$ is an invertible function.
\end{lemma}

\begin{proof}
(a) Lemma \ref{NS-om-lem} immediately shows that the map
$$\OO_S\to \Pi\om_{X/S}: f\mapsto f\cdot (s_1+s_2)$$
is injective.
Furthermore, the submodule $\OO_X(s_1+s_2)$ has the following topological basis over $\OO_S$:
\begin{align*}
&s_1+s_2, \ \ z_1^{n+1}(s_1+s_2)=z_1^n(z_1+t)s_1, \ \
z_2^n(z_2-t)s_2, \ \ \th_1(s_1+s_2)=\th_1s_1+ts_0, \\
&z_1^{n+1}\th_1(s_1+s_2)=z_1^n(z_1+t)\th_1s_1, \ \
\th_2(s_1+s_2)=\th_2s_2+ts_0, \ \ z_2^n(z_2-t)\th_2s_2,
\end{align*}
where $n\ge 0$.
Hence, the quotient by this submodule has the images of $s_0, s_1$ as a basis over $\OO_S$.

\noindent
(b) Near the node, $\om_{X/S}^2$ is a free $\OO_X$-module with one even generator $e$. Furthermore, it
is easy to check that with respect to the map $\om_{X/S}^{\ot 2}\to \om_{X/S}^2$ one has
$$s_1^2=z_1e, \ \ s_2^2=-z_2e, \ \ s_1s_2=te, \ \ s_0s_i=\th_ie,$$
for $i=1,2$.
Thus, the image of the multiplication by $s_1+s_2$ is the $\OO_X$-submodule generated
by
$$s_1(s_1+s_2)=(z_1+t)e, \ \ s_2(s_1+s_2)=(t-z_2)e, \ \ s_0(s_1+s_2)=(\th_1+\th_2)e.$$
In other words, we have an identification
$$\hat{\GG}\simeq \OO_X/(z_1+t, z_2-t, \th_1+\th_2)=\OO_Z\simeq \OO_S[\th_1],$$
as claimed.

\noindent
(c) The operator of multiplication by $s_1$ acts on the bases of $\hat{\FF}$ and $\hat{\GG}$ as
$$s_1\mapsto s_1^2=-te, \ \ s_0\mapsto s_1s_0=\th_1e,$$
Hence, the Berezinian is equal to $-t$ (recall that $s_1$ is an even generator of $\hat{\FF}$, while $s_0$ is an odd generator).
\end{proof}

Let us consider the following resolutions for the derived push-forwards under $\pi$:
\begin{align*}
&R\pi_*(\OO_X): & [\pi_*(\Pi \om_{X/S}(D)\to \pi_*\FF], \\
&R\pi_*(\Pi \om_{X/S}): & [\pi_*(\om_{X/S}^2(D))\to \pi_*\GG], \\
&R\pi_*(\Pi \om_{X/S}): & [\pi_*(\Pi \om_{X/S}(D))\to \pi_*(\Pi \om_{X/S}(D)|_D)], \\
&R\pi_*(\om_{X/S}^2): & [\pi_*(\om_{X/S}^2(D))\to \pi_*(\om_{X/S}^2(D)|_D)].
\end{align*}
These resolutions give a canonical isomorphism over $S$,
\begin{equation}\label{Ber-om2-NS-node-eq}
\BB(\om_{X/S}^2)\ot \BB(\OO_X)\simeq\Ber(R\pi_*(\Pi\om_{X/S}))^{\ot 2}\ot \Ber(\pi_*\GG)\ot \Ber(\pi_*\FF)^{-1}.
\end{equation}
Now the super Mumford isomorphism for the induced smooth family
$X_{S'}\to S'$ is obtained by choosing an isomorphism of
$$\FF|_{X_{S'}}\simeq \GG|_{X_{S'}}$$
(where both sheaves are supported on the zero divisor of $s$ in $X_{S'}$)
and using the Berezinian of the induced map on $R\pi_*$ to get an isomorphism of $\Ber(\pi_*\FF)|_{S'}$ with $\Ber(\pi_*\GG)|_{S'}$ (see \eqref{Ber-L-D-eq}).
For this we are going to construct a morphism
$$\a:\FF\to \GG,$$
which is an isomorphism away from the node, and use its restriction to $X_{S'}$.

Note that both $\FF$ and $\GG$ are supported on $Z\sqcup E$,
where $E$ is a relative divisor supported in the smooth locus of $\pi$, so
we have decompositions into subsheaves supported on $Z$ and on $E$,
$$\FF\simeq \FF_Z\oplus \FF_{E}, \ \ \GG\simeq \GG_Z\oplus \GG_{E}.$$
We can choose separately morphisms $\FF_{E}\to \GG_{E}$ and $\FF_Z\to \GG_Z$.

We have
$$\FF_{E}\simeq \pi_*(\Pi \om_{X/S}(D)|_{E}), \ \ \GG_{E}\simeq \pi_*(\om_{X/S}^2(D)|_{E}),$$
so both are isomorphic to $\OO_{E}$, and we choose any isomorphism between them as a morphism $\a_{E}:\FF_{E}\to \GG_{E}$.
On the other hand, we have
$$\FF_Z\simeq \hat{F}, \ \ \GG_Z\simeq \hat{\GG},$$

Hence, we can use the morphism of $\OO_X$-modules,
$$\mu_{s_1}:\hat{\FF}\to \hat{\GG},$$
given by the multiplication with $s_1$, as our morphism $\FF_Z\to \GG_Z$.
By Lemma \ref{F-G-completions-lem}, it restricts to an isomorphism over $t\neq 0$.
Thus, we get the desired morphism
$$\a=(\mu_{s'},\a_{E}):\FF\to \GG.$$

By definition, the super Mumford isomorphism for $X_{S'}\to S'$ is obtained by restricting the isomorphism of ilne bundles
\eqref{Ber-om2-NS-node-eq} to $S'$ and multiplying it with $\ber R\pi_*(\a|_{X_{S'}})^{-1}\in \BB(\FF)\ot \BB(\GG)^{-1}$.
Since $\ber R\pi_*(\a_{E'})$ is invertible on $S$, we see that
$$\ber R\pi_*(\a|_{X_{S'}})=u\cdot \ber \pi_*(\mu_{s_1}),$$
where $u$ is an invertible function on $S$.

Using Lemma \ref{F-G-completions-lem}(iii), we deduce that
$$\ber R\pi_*(\a|_{X_{S'}})^{-1}=u'\cdot t^{-1},$$
for some invertible function $u'$ on $S$, with respect to some
bases of $\BB(\FF)$ and $\BB(\GG)$ on $S$.

Thus, we obtained an isomorphism
\begin{equation}\label{partial-Mumford-isom}
\BB(\om_{X/S}^2)\rTo{\sim} \BB(\OO_X)^{-1}\ot \Ber(R\pi_*(\Pi\om_{X/S}))^{\ot 2}(\De),
\end{equation}
where $\De$ is the divisor given by $t=0$. Combining this with isomorphism
\eqref{BB-om3-X-Mum-eq}, we get the following local result.

\begin{prop}\label{super-Mum-prop}
In the above situation the super Mumford isomorphism
$$\BB(\om_{X'/S'}^{-2})\simeq \BB(\om_{X'/S'}^3)\simeq \BB(\OO_{X'})^5$$
extends to an isomorphism
$$\BB(\om_{X/S}^{-2})\simeq \BB(\om_{X/S}^3)\simeq \BB(\OO_X)^5(-\De).$$
\end{prop}

Combining this with Corollary \ref{KS-super-NS-node-cor} and with Proposition \ref{KS-super-R-prop} we obtain the following result.

\begin{theorem}\label{canonical-class-no-punct-thm}
Let $\ov{\SS}_g$ denote the moduli stack of stable supercurves of genus $g$, and
let $\De_{NS}\sub \ov{\SS}_g$ (resp., $\De_R\sub \ov{\SS}_g$) denote the boundary divisor supported on the locus where a supercurve
acquires an NS node (resp., a Ramond node), defined in Sec.\ \ref{boundary-equation-sec}.
Then one has an isomorphism of line bundles
\begin{equation}\label{can-class-smooth-moduli-eq}
K_{\ov{\SS}_g}\simeq \Ber_1^5(-2\De_{NS}-\De_R).
\end{equation}

\end{theorem}

\begin{proof}
Let $\pi:X\to\ov{\SS}_g$ denote the universal supercurve.
First, we have an isomorphism
$$K_{\ov{\SS}_g}\simeq \Ber^{-1} R^1\pi_*\om_{X/\ov{\SS}_g}^{-2}(-\De_{NS}-\De_R)\simeq \BB(\om_{X/\ov{\SS}_g}^{-2})(-\De_{NS}-\De_R)$$
(see Corollary \ref{KS-super-NS-node-cor} and Proposition \ref{KS-super-R-prop}).
Next, by Proposition \ref{super-Mum-prop}, we have
$$\BB(\om_{X/\ov{\SS}_g}^{-2})\simeq \BB(\om_{X/\ov{\SS}_g}^3)\simeq \Ber_1^5(-\De_{NS})$$
(recall that the super-Mumford isomorphism \eqref{super-Mumford-eq} extends over the Ramond type boundary divisor $\De_R$).
Combining this with the previous isomorphism we get the result.
\end{proof}

\subsection{Canonical line bundle on the moduli space of stable supercurves with punctures}\label{can-line-bun-punctures-sec}

Let $\pi:X\to S$ be a stable supercurve with NS punctures $P_1,\ldots,P_m$ and Ramond divisors $R_1,\ldots,R_n$.
We denote by

$D_i\sub X$ the divisors associated with $P_i$ as in Sec.\ \ref{NS-correspondence-sec}.

For every $i=1,\ldots,m$, let us define a line bundle of rank $0|1$ on $S$,
$$\Psi_i:=P_i^*\om_{X/S}.$$

\begin{lemma}\label{psi-lemma}
One has natural isomorphisms of line bundles of rank $0|1$ on $S$,
$$\Psi_i\simeq \pi_*\OO_{D_i}/\OO_S,$$
$$\Psi_i^{-1}\simeq P_i^*\om_{X/S}(D_i)\simeq P_i^*\om_{D_i/S}\simeq \Ber(\pi_*\OO_{D_i}).$$
Hence, for any line bundle $L$ of rank $1|0$ on $X$, one has
$$\Ber(\pi_*L|_{D_i})\simeq \Psi_i^{-1},$$
while for a line bundle $M$ of rank $0|1$, one has
$$\Ber(\pi_*M|_{D_i})\simeq \Psi_i.$$
\end{lemma}

\begin{proof}
Let us set $\Psi=\Psi_i$ and
$$\wt{\Psi}:=\pi_*\OO_D/\OO_S,$$
where $D=D_i$ and $P=P_i$.
Then $\wt{\Psi}$ is a line bundle of rank $0|1$, and we have
$$\Ber(\pi_*\OO_D)\simeq \wt{\Psi}^{-1}.$$
Hence, from \eqref{Ber-L-D-eq} we get the last two isomorphisms with $\Psi_i$ replaced by $\wt{\Psi}$.

Applying this to the line bundle $\om_{X/S}^{-2}$ of rank $1|0$, we get an isomorphism
$$\Ber(\pi_*\om_{X/S}^{-2}|_D)\simeq \wt{\Psi}^{-1}.$$
But from Lemma \ref{NS-div-corr-lem}, we have an isomorphism over $S$,
$$\pi_*(\om_{X/S}^{-2}|_D)\simeq \TT_{X/S}|_P.$$
Hence, passing to the Berezinians we get
$$\wt{\Psi}^{-1}\simeq \om_{X/S}^{-1}|_P,$$
i.e., $\wt{\Psi}\simeq \Psi$.

Next, let $I_D\sub I_P\sub \OO_X$ be the ideal sheaves of $D=D_i$ and $P=P_i$. Since the projection to $S$ induces an isomorphism $P\simeq S$,
we have a decomposition
$$\pi_*\OO_D=\OO_S\oplus \pi_*(I_P/I_D),$$
so
$$\Psi\simeq \wt{\Psi}\simeq \pi_*(I_P/I_D).$$
Note also that $I_P^2\sub I_D$, so $I_P/I_D$ can be identified with the conormal sheaf to $P$ in $D$.
Now, since $\om_{P/S}=\OO_S$, the
exact sequence
$$0\to \Psi \to \Om_{D/S}|_P\to \OO_P\to 0$$
gives an isomorphism
$$P^*\om_{D/S}\simeq \Ber(\Psi)=\Psi^{-1}.$$
\end{proof}

\begin{cor}\label{square-NS-triv-cor}
The line bundle $P_i^*\om_{X/S}^2(D_i)$ on $S$ is canonically trivialized.
\end{cor}

\begin{example}
Suppose $X\to S$, $(P_i)$, is a family of supercurves with NS-punctures over an {\it even} base $S$.
Then $\OO_X=\OO_C\oplus L$, where $(C,L)$ is the underlying family of curves with spin-structures, and
$\II_{P_i}=\II_{p_i}\oplus L$, where $p_i\sub C$ are marked points on $C$. We also have $\II_D=\II_{p_i}\ot_{\OO_C} \OO_X$.
Thus,
$$\Psi_i^{-1}=P_i^*(\om_{X/S}(D_i))=P_i^*(L(p_i)\oplus \om_{C/S}(p_i))\simeq p_i^*L(p_i)\simeq p_i^*L^{-1},$$
where the last isomorphisms is induced by the trivialization of $L^2(p_i)|_{p_i}\simeq \om_{C/S}(p_i)|_{p_i}$.
\end{example}

\begin{remark}\label{Res-rem}
Note that for an NS-puncture $P_i$ and the corresponding divisor $D_i$, the decomposition
\eqref{NS-div-splitting-eq} can be rewritten as
$$\pi_*\OO_{D_i}=\OO_S\oplus \Psi_i,$$
where $\Psi_i$ is a square-zero ideal.
For any line bundle $L$ over $D_i$ we have an exact sequence
$$0\to P_{i_*}(\Psi_i\ot P_i^*L)\to L\to P_{i*}P_i^*L\to 0$$

In the case $L=\om_{X/S}(D_i)|_{D_i}=\om_{D_i/S}$, the induced exact sequence of push-forwards to $S$ has a splitting
$$\pi_*(\om_{X/S}(D_i)/\om_{X/S})\to \OO_S$$
given by the residue map for the $0|1$-dimensional superscheme $D_i/S$, so we have a decomposition
$$\pi_*(\om_{D_i/S})=\Psi_i^{-1}\oplus \OO_S.$$
\end{remark}

For every Ramond divisor $R_j\sub X$ let us consider the line bundle on the base,
$$\Phi_j:=\Ber(\pi_*\OO_{R_j})^{-1}\simeq \Ber(\pi_*\om_{R_j}).$$
Note that $\Phi_j=\Phi_{R_j}$ (see \eqref{PhiR-def}), so the line bundles $\Phi_j^2$ are canonically trivialized.

Let us consider the line bundle of rank $1|0$ over $X$,
$$\LL(X,P_\bullet,R_\bullet):=\om_{X/S}^{-2}(-\sum_i D_i-2\sum_j R_j).$$

\begin{lemma}\label{Psi-Phi-contribution-lem}
One has a natural isomorphism
$$\BB(\LL(X,P_\bullet,R_\bullet))\simeq \BB(\om_{X/S}^3)\ot\bigotimes_i\Psi_i$$

\end{lemma}

\begin{proof}
Let us set $D=\sum_i D_i$, $R=\sum_j R_j$.
First, the Grothendieck duality gives an isomorphism
$$\BB(\LL(X,P_\bullet,R_\bullet))\simeq \BB(\om_{X/S}^3(D+2R)).$$
Now using the exact sequence
\begin{equation}\label{om-3-sh-ex-seq}
0\to \om_{X/S}^3(2R)\to \om_{X/S}^3(2R)(D)\to \om_{X/S}^3(2R)(D)|_D\to 0,
\end{equation}
we get using \eqref{Ber-L-D-eq} and Lemma \ref{psi-lemma},
$$\BB(\om_{X/S}^3(D+2R)) \simeq \BB(\om_{X/S}^3(2R))\ot \Ber(\pi_*\om_{X/\MM}^3(D)|_D)\simeq
\BB(\om_{X/M}^3(2R))\ot\bigotimes_i\Psi_i.$$
Similarly, using the fitration of $\om_{X/S}^3(2R)/\om_{X/S}^3$ with subquotients
$\om_{X/S}^3(2R)|_R$ and $\om_{X/S}^3(R)|_R$, we get an isomorphism
$$\BB(\om_{X/S}^3(2R))\simeq \BB(\om_{X/S}^3)\ot\bigotimes_j\Phi_j^2.$$
It remains to we use the trivializations of $\Phi_j^2$.
\end{proof}

Now the same argument as in Theorem \ref{canonical-class-no-punct-thm} gives the formula for the canonical line bundle on the moduli stack of stable supercurves with punctures.

\begin{proof}[Proof of Theorem B]
Let $\ov{\SS}=\ov{\SS}_{g,n_{NS},n_R}$. We combine the isomorphism
$$K_{\ov{\SS}}\simeq \Ber^{-1} R^1\pi_*\LL(X,P_\bullet,R_\bullet)(-\De_{NS}-\De_R)\simeq \BB(\LL(X,P_\bullet,R_\bullet))(-\De_{NS}-\De_R)$$
(see Sec.\ \ref{Kodaira-punct-sec}) with Lemma \ref{Psi-Phi-contribution-lem} and the isomophism
$$\BB(\om_{X/\ov{\SS}}^3)\simeq \Ber_1^5(-\De_{NS}).$$
This gives the required isomorphism
\begin{equation}\label{canonical-class-eq}
K_{\ov{\SS}}\simeq \Ber_1^5\otimes \bigotimes_{i=1}^m\Psi_i(-2\De_{NS}-\De_R),
\end{equation}
\end{proof}

\section{Splitting at the boundary divisor}
\label{Splitting-sec}

Now we are going to study the restriction of the isomorphism of Theorem B to the boundary divisor. Using our presentation of the line bundle corresponding
to the boundary divisor as a Berezinian (see Sec.\ \ref{boundary-equation-sec}) we find a natural identification of the normal line bundle to the boundary divisor.
Then we give a proof of Theorem C concerning the NS boundary component. We also give a conjectural statement for the Ramond boundary component.

\subsection{NS boundary components}\label{NS-splitting-sec}

Let $\iota:B\to \ov{\SS}$ be the standard gluing map covering an NS type boundary component, i.e., one of the maps \eqref{NS-sep-node-morphism} or
\eqref{NS-nonsep-node-morphism}, restricted to the locus of smooth supercurves.
Let $X_B\to B$ denote the universal stable supercurve, which is obtained by identifying two NS punctures $P_1,P_2$ on a smooth supercurve
$\wt{X}\to B$ into a node $Q\sub X_B$. Let $D_1,D_2\sub \wt{X}$ be the corresponding divisors. Note that we have a finite morphism $\rho:\wt{X}\to X_B$, and an exact sequence
on $X_B$,
$$0\to \OO_{X_B}\to \rho_*\OO_{\wt{X}}\to \OO_Q\to 0.$$
In particular, the Berezinian line bundle $\BB(\OO_{X_B})$ for the family $X_B\to B$ is naturally identified with $\BB(\OO_{\wt{X}})$ defined for the family $\wt{X}\to B$.

Note that in the case of a separating node, where $B=\SS_1\times \SS_2$, the line bundle $\BB(\OO_{\wt{X}})$ is the exterior product of two similar line bundles on the factors
$\SS_1$ and $\SS_2$.

We can rewrite \eqref{canonical-class-eq} near $B$ as an isomorphism
$$K_{\ov{\SS}}(2\De_{NS})\simeq \BB(\OO_X)^5.$$
Thus, pulling it back to $B$ leads to an isomorphism
$$K_B\ot N_B\simeq \BB(\OO_{X_B})^5,$$
where $N_B$ is the normal bundle defined as the pull-back of $\OO(\De_{NS})$ to $B$.
Note that the universal supercurve over $B$ is equipped with two NS punctures $P_1,P_2$ and the isomorphism \eqref{canonical-class-eq}
in smaller genus gives
$$K_B\simeq \BB(\OO_{\wt{X}})^5\ot\Psi_1\ot\Psi_2.$$
Comparing with the previous isomorphism we get an isomorphism

$$N_B\simeq \Psi_1^{-1}\ot \Psi_2^{-1}.$$

Below we will define such a canonical isomorphism independently and then will check its compatibility with two above isomorphisms.

First, recall that we have a line bundle $\om_{X/S}^2$ defined on the universal curve of $\ov{\SS}$ by extending from the smooth locus (see Theorem \ref{can-square-thm}).

\begin{lemma}\label{splitting-ex-seq-lem}
(i) For any integer $m$ let us set
$$\om_{X_B/B}^{2m}:=\om_{X/\ov{\SS}}^{2m}|_B.$$
Then one has an exact sequence on $X_B$,
\begin{equation}\label{om-2m-res-ex-seq}
0\to \om_{X_B/B}^{2m}\to \rho_*\om_{\wt{X}/B}^{2m}(mD_1+mD_2)\to \OO_Q\to 0
\end{equation}
where $Q\sub X_B$ is the relative node obtained by gluing $P_1\simeq B\simeq P_2$.
Here we use the canonical trivializations of the restrictions of $\om_{\wt{X}/B}^2(D_1+D_2)$ to $P_1$ and $P_2$
(see Corollary \ref{square-NS-triv-cor}).
Hence, we have a natural isomorphism
\begin{equation}\label{om-2m-res-NS-eq}
\BB(\om_{X/\ov{\SS}}^{2m})|_B\simeq \BB(\om_{\wt{X}/B}^{2m}(mD_1+mD_2)).
\end{equation}

\noindent
(ii) Let us set $\om_{X_B/B}^{2m+1}:=\om_{X_B/B}^{2m}\ot \om_{X_B/B}$. Then one has an exact sequence on $X_B$,
\begin{equation}\label{om-res-simple-seq}
0\to \rho_*\om^{2m+1}_{\wt{X}/B}(mD_1+mD_2)\to \om_{X_B/B}^{2m+1}\to Q_*\OO_S\to 0.
\end{equation}
Hence, we have a natural isomorphism
\begin{equation}\label{om-2m+1-res-NS-eq}
\BB(\om_{X/\ov{\SS}}^{2m+1})|_B

\simeq\BB(\om_{\wt{X}/B}^{2m+1}(mD_1+mD_2)).
\end{equation}

In addition, we have a natural exact sequence
$$0\to \om_{X_B/B}^{2m+1}\to \rho_*\om_{\wt{X}/B}^{2m+1}((m+1)D_1+(m+1)D_2)\to \CC\to 0,$$
where $\CC$ is a sheaf supported on the node fitting into an exact sequence
\begin{equation}\label{Q-C-Psi-seq}
0\to Q_*\OO_S\to \CC\to Q_*(\Psi_1^{-1}\oplus \Psi_2^{-1})\to 0.
\end{equation}

\noindent
(iii) One has an exact sequence on $X_B$,
$$0\to \KK\to \Om_{X_B/B}\to \rho_*\Om_{\wt{X}/B}\to 0,$$
where the sheaf $\KK$ is supported on the node, and has a filtration with the subfactors
$$Q_*(\Psi_1^2\Psi_2^2), \ Q_*(\Psi_1^2\Psi_2), \ Q_*(\Psi_1\Psi_2^2), \ Q_*(\Psi_1\Psi_2).$$
In particular, $\Ber \pi_*\KK$ is canonically trivial,
so
$$\Ber R\pi_*(\Om_{X_B/B})\simeq\Ber R\pi_*\Om_{\wt{X}/B}.$$

\noindent
(iv) One has an exact sequence over the smooth locus of $B$,
$$0\to j_*\Om_{U/\ov{\SS}}|_{X_B}\to \Om_{\wt{X}/B}(D_1+D_2)\to \wt{\CC}\to 0,$$
where $U\sub X$ is the smooth locus of $X\to \ov{\SS}$,
and the sheaf $\wt{\CC}$ is supported on the node and has a filtration with the subfactors
$$\OO_Q,  \ \OO_Q, \  Q_*(\Psi_1^{-1}\oplus \Psi_2^{-1}).$$
\end{lemma}

\begin{proof}
(i) We just have to identify the pull-back of the line bundle $\om_{X_B/B}^2$ to $\wt{X}$, $\rho^*\om_{X_B/B}^2$ with
$\om_{\wt{X}/B}^2(D_1+D_2)$. Note that we have a natural identification of these line bundles over the smooth locus of $\wt{X}$.
Thus, we need to check that it extends to an isomorphism over the node. For this, it is enough to study these line bundle in the formal
neighborhood of the node. Thus, we can place ourselves in the framework of Sec.\ \ref{super-KS-NS-sec} and use a generator
$e$ of $\om_{X/S}^2$ (see \eqref{e-sec-eq}). Since $\rho^*e$ is a generator of $\om_{\wt{X}/B}^2(D_1+D_2)$, the assertion follows.

\noindent
(ii) First, let us consider the case $m=0$.
Using Lemma \ref{NS-om-lem} we easily see that there is an injective map $\om_{X_B/B}\to \rho_*\om_{\wt{X}/B}(D_1+D_2)$,
and that $\om_{X_B/B}$ contains $\rho_*\om_{\wt{X}/B}$. Thus, we have
$$\CC:=\coker(\om_{X_B/B}\to \rho_*\om_{\wt{X}/B}(D_1+D_2))=\coker(\om_{X_B/B}/\om_{\wt{X}/B}\to \rho_*(\om_{\wt{X}/B}(D_1+D_2)/\om_{\wt{X}/B})).$$

Note that since $\Psi_i^{-1}\simeq P_i^*\om_{\wt{X}/B}(D_i)$,
by Remark \ref{Res-rem}, we have an exact sequence
$$0\to \OO_{P_1}\oplus \OO_{P_2}\to \om_{\wt{X}/B}(D_1+D_2)/\om_{\wt{X}/B}\to P_{1*}\Psi_1^{-1}\oplus P_{2*}\Psi_2^{-1}\to 0$$
In local coordinates, the projection to $P_{i*}\Psi_i^{-1}$ sends $\frac{1}{z_i}[dz_i|d\th_i]$ to a generator, and sends $\frac{\th_i}{z_i}[dz_i|d\th_i]$ to zero.
In particular, this projection vanishes on the image of $\om_{X_B/B}/\om_{\wt{X}/B}$.
Thus, we have an embedding

$$\om_{X_B/B}/\om_{\wt{X}/B}\hra \OO_Q\oplus \OO_Q.$$
We claim that $\om_{X_B/B}/\om_{\wt{X}/B}$ coincides with the kernel of
the addition map $\OO_Q\oplus \OO_Q\to \OO_Q$.
Indeed, this is a local statement. In local coordinates the generators of two summands $\OO_Q$ are $\frac{\th_i}{z_i}[dz_i|d\th_i]$,
and the generator $s_0$ of $\om_{X_B/B}/\om_{\wt{X}/B}$ is mapped to their difference.

From this we also see that $\CC$ fits into the exact sequence \eqref{Q-C-Psi-seq}.

To derive the case of arbitrary $m$ from that of $m=0$, we tensor the sequence \eqref{Q-C-Psi-seq} with the line bundle
$\om^{2m}_{X_B/B}$. By the triviality of $Q^*(\om_{X_B/B}^2$, we get the sequence of the required form.

\noindent
(iii) First, we note that $\KK$ is supported on the node, so it is enough to prove the assertion after replacing $X_B$ with the formal neighborhood of the node.
Then $\wt{X}$ becomes the union of two branches $X_1\sqcup X_2$, so that $P_1\in X_1$, $P_2\in X_2$.
Now we have inclusions of ideals
$$\rho_*(I_{P_1}\oplus 0), \rho_*(0\oplus I_{P_2})\sub \OO_{X_B}\sub \rho_*\OO_{\wt{X}}.$$
Furthermore, the product of these ideals is zero. This implies that for $a_1\in I_{P_1}$, $a_2\in I_{P_2}$, one has
$$(0,a_2)\cdot d(a_1,0)=\pm (a_1,0)\cdot d(0,a_2).$$
Hence, we have well defined map
$$\kappa:I_{P_1}/I_{P_1}^2\ot I_{P_2}/I_{P_2}^2\to \KK\sub\Om_{X_B/B}: a_1\ot a_2)\mapsto (a_1,0)\cdot d(0,a_2).$$

Note that $I_{P_1}/I_{P_1}^2$ fits into an exact sequence
$$0\to I_{D_1}/I_{D_1}I_{P_1}\to I_{P_1}/I_{P_1}^2\to I_{P_1}/I_{D_1}\to 0$$
with $I_{D_1}/I_{D_1}I_{P_1}\simeq \Psi_1^2$ and $I_{P_1}/I_{D_1}\simeq \Psi_1$.
Using Lemma \ref{differential-generators-lem}(ii), one checks that
$\kappa$ is an isomorphism, and the assertion follows.

\noindent
(iv) This follows from parts (i), (ii) and from the exact sequence \eqref{j-Om-ex-seq} (and a similar exact sequence for $\Om_{\wt{X}/B}$ which
holds over the smooth locus).
\end{proof}

We can use the definition of the NS boundary line bundle \eqref{boundary-line-bundle} to compute the normal bundle
over the smooth part of $B$.
From the above Lemma we get
$$\Ber R\pi_*(j_*\Om_{U/S})|_B\simeq \Ber R\pi_*(\Om_{\wt{X}/B}(D_1+D_2))\ot \Psi_1^{-1}\Psi_2^{-1},$$
$$\Ber R\pi_*(\Om_{X/S}|_B)\simeq \Ber R\pi_* \Om_{\wt{X}/B}.$$
Finally the exact sequence
$$0\to \om_{\wt{X}/B}^2\to \Om_{\wt{X}/B}\to \om_{\wt{X}/B}\to 0$$
near $D_1,D_2$, together with Lemma \ref{psi-lemma}, show that
$$\Ber R\pi_* \Om_{\wt{X}/B}(D_1+D_2)\simeq \Ber R\pi_*\Om_{\wt{X}/B}.$$
Hence, we deduce an isomorphism
\begin{equation}\label{NS-normal-bundle-formula}
\OO(\De)|_B\simeq \Ber R\pi_*(j_*\Om_{U/S})|_B\ot \Ber^{-1} R\pi_*(\Om_{X/S}|_B)\simeq \Psi_1^{-1}\Psi_2^{-1}.
\end{equation}

Now we can state a compatibility result between the super Mumford isomorphisms over $\ov{\SS}$ and over the NS boundary component $B$.
Recall that we have the super Mumford isomorphism
$$\mu_{\ov{\SS}}:\BB(\om_{X/\ov{\SS}}^{-2})\rTo \BB(\OO_X)^5(-\De)$$
near $B$ (see Prop.\ \ref{super-Mum-prop}). We also have the super Mumford isomorphism for the family $\wt{X}/B$, over the smooth locus:
$$\mu_B:\BB(\om_{\wt{X}/B}^{-2}(-D_1-D_2))\rTo{\sim} \BB(\OO_{\wt{X}})^5\ot \Psi_1\ot\Psi_2$$
(see Lemma \ref{Psi-Phi-contribution-lem}).

\begin{theorem}\label{splitting-thm}
The following diagram of isomorphisms of line bundles on $B$ is commutative up to a sign
\begin{diagram}
\BB(\om_{X/\ov{\SS}}^{-2})|_B&\rTo{\mu_{\ov{\SS}}|_B}&\BB(\OO_X)^5(-\De)|_B\\
\dTo{\eqref{om-2m-res-NS-eq}}&&\dTo{\eqref{NS-normal-bundle-formula}}\\
\BB(\om_{\wt{X}/B}^{-2}(-D_1-D_2))&\rTo{\mu_B}&\BB(\OO_{\wt{X}})^5\ot \Psi_1\ot\Psi_2
\end{diagram}

\end{theorem}

Recall that to get $\mu_{\ov{\SS}}$ we used three isomorphisms: the Grothendieck-Serre duality isomoprhism
$$SD_{X/\ov{\SS}}:\BB(\om_{X/\ov{\SS}}^{-2})\rTo{\sim}\BB(\om_{X/\ov{\SS}}^3),$$
the isomorphism
$$\BB(\om_{X/\ov{\SS}}^3)\simeq \BB(\om_{X/\ov{\SS}}^2)^{-1}\ot \BB(\OO_X)^2,$$
and the isomorphism
$$\BB(\om_{X/\ov{\SS}}^2)\simeq \BB(\om_{X/\ov{\SS}})^{-2}\ot \BB(\OO_X)^{-1}(\De)$$
(see Sec.\ \ref{super-Mum-NS-sec}). Similarly, $\mu_B$ is a composition of similar three isomorphisms. So we can reduce the proof of Theorem \ref{splitting-thm}
to separate compatibilities involving each of these three isomorphisms. We deal with this compatibilities in the next three lemmas.

\begin{lemma}\label{split-SD-lem}
For any $m\in\Z$, the following diagram of isomorphisms between line bundles on $B$ is commutative:
\begin{diagram}
\BB(\om_{X/\ov{\SS}}^{-2m})|_B&\rTo{SD_{X/\ov{\SS}}|_B}&\BB(\om_{X/\ov{\SS}}^{2m+1})|_B\\
\dTo{\eqref{om-2m-res-NS-eq}}&&\dTo{\eqref{om-2m+1-res-NS-eq}}\\
\BB(\om_{\wt{X}/B}^{-2m}(-mD_1-mD_2))&\rTo{SD_{\wt{X}/B}}&\BB(\om_{\wt{X}/B}^{2m+1}(mD_1+mD_2))
\end{diagram}
where the horizontal arrows are given by Grothendieck-Serre duality.

\end{lemma}

\begin{proof}
To begin with we can replace the upper horizontal arrow with the one induced by the Grothendieck-Serre duality for $X_B$ over $B$,
$$\BB(\om^{-2m}_{X_B/B})\rTo{SD_{X_B/B}}\BB(\om^{2m+1}_{X_B/B}).$$
Let us set for brevity
$$L:=\om^{-2m}_{X_B/B}, \ \ \wt{L}:=\om_{\wt{X}/B}^{-2m}(-mD_1-mD_2).$$

Recall that for a sufficiently nice morphism $f:X\to Y$ and a perfect complex $F$ on $X$, the Grothendieck Serre duality gives an isomorphism
$$SD_f:Rf_*(F)\rTo{\sim} Rf_*(R\und{\Hom}(F,\om_f[\dim f]))^\vee.$$

Then we claim that there is an isomorphism of the exact triangles induced by the exact sequences \eqref{om-2m-res-ex-seq} and \eqref{om-res-simple-seq}
\begin{equation}\label{morphism-SD-triangles-eq}
\begin{diagram}
R\pi_*(L)&\rTo{}&R\pi_*(\rho_*\wt{L})&\rTo{}&\OO_S&\rTo{}\ldots\\
\dTo{SD_{X_B/B}}&&\dTo{SD_{\wt{X}/B}}&&\dTo{\id}\\
R\pi_*(L^{-1}\ot\om_{X_B/B}[1])^\vee&\rTo{}&R\pi_*(\rho_*(\wt{L}^{-1}\ot\om_{\wt{X}/B}[1]))^\vee&\rTo{}&\OO_S&\rTo{}\ldots
\end{diagram}
\end{equation}
Clearly this would imply the claimed commutativity.

Now we claim that commutativity of both squares in \eqref{morphism-SD-triangles-eq} follows from the
general property of Grothendieck-Serre duality for a pair of morphisms $X\rTo{g}Y\rTo{f}X$,
\begin{diagram}
Rf_*(Rg_*(F))&\rTo{\sim}&R(f\circ g)_*(F)\\
\dTo{SD_f}&&\dTo{SD_g}\\
Rf_*(R\und{\Hom}(Rg_*(F),\om_f[\dim f]))^\vee&\rTo{\sim}&R(f\circ g)_*(R\und{\Hom}(F,\om_{f\circ g}[\dim (f\circ g)]))^\vee
\end{diagram}
where the lower horizontal arrow is induced by the Grothendieck-Serre duality isomorphism
$$R\und{\Hom}(Rg_*(F),\om_f[\dim f])\simeq Rg_*R\und{\Hom}(F,\om_{f\circ g}[\dim (f\circ g)]).$$

Indeed, applying this to $f=\pi$, $g=\rho$ and $F=\wt{L}$ allows us to identify the middle vertical arrow in \eqref{morphism-SD-triangles-eq} with
the map
$$R\pi_*(\rho_*\wt{L})\to R\pi_*R\und{\Hom}(\rho_*\wt{L},\om_{X_B/B}[1])^\vee$$
given by the Grothendieck-Serre duality for $X_B/B$. Taking this into account, commutativity of the left square in \eqref{morphism-SD-triangles-eq}
becomes a basic functoriality of $SD_{\pi}$. On the other hand, commutativity of the right square in \eqref{morphism-SD-triangles-eq} similarly follows
from the functoriality of $SD_{\pi\circ \rho}$ applied to the natural morphism $\wt{L}\to Q_*\OO_S$ and
from the above compatibility for $f=\pi\circ \rho$, $g=Q$ and $F=\OO_S$.
\end{proof}

\begin{lemma}\label{resolution-Ber-compatibility-lem}
Let $\rho:Y\to X$ be a morphism of families of stable supercurves over $B$, which is a fiberwise resolution of the node $Q:B\to X$ (so it is an isomorphism
away from $X\setminus Q(B)$). For any coherent sheaf $\FF$ on $Y$ which is locally free of rank $1|0$ over the smooth locus, and any
line bundle $L$ on $X$ with a trivialization of $Q^*L$, we have a commutative diagram of isomorphisms of line bundles on $B$,
\begin{diagram}
\BB(\rho_*\FF\ot L)&\rTo{\eqref{super-Deligne-sheaf-eq}}&\BB(\rho_*\FF)\ot \BB(L)\ot \BB(\OO_X)^{-1}\\
\dTo{}&&\dTo{}\\
\BB(\FF\ot \rho^*L)&\rTo{\eqref{super-Deligne-sheaf-eq}}&\BB(\FF)\ot \BB(\rho^*L)\ot \BB(\OO_Y)^{-1}
\end{diagram}
where in the right vertical arrow we use isomorphisms $\BB(\OO_X)\simeq \BB(\OO_Y)$ and $\BB(L)\simeq \BB(\rho^*L)$
coming from the exact sequences
$$0\to \OO_X\to \rho_*\OO_Y\to \OO_Q\to 0$$
$$0\to L\to L\ot \rho_*\OO_Y\to L\ot \OO_Q\to 0$$
and the trivialization of $Q^*L$.
\end{lemma}

\begin{proof}
The question is local in the base, so we can assume that we can
choose a relative divisor $D\sub X$ supported in the smooth locus of $\pi:X\to B$, and a global section $s$ of $L(D)$ such that the
$E=\div(s)$ is also supported in the smooth locus. Now we compute both horizontal arrows using the section $s$ on $X$ and its pull-back $\rho^*s$ on $Y$.
Thus, for the top horizontal arrow we use resolutions
\begin{align*}
&R\pi_*(\OO_X): & [\pi_*L(D)\to \pi_*(L(D)|_E)], \\
&R\pi_*(\rho_*\FF): & [\pi_*(\rho_*\FF\ot L(D))\to \pi_*(\rho_*\FF\ot L(D)|_E)], \\
&R\pi_*(L): & [\pi_*(L(D))\to \pi_*(L(D)|_D)], \\
&R\pi_*(\rho_*\FF\ot L): & [\pi_*(\rho_*\FF\ot L(D))\to \pi_*(\rho_*\FF \ot L(D)|_D)],
\end{align*}
while for the bottom horizontal arrow we use similar resolutions on $Y$ that use the section $\rho^*s$ of $\rho^*L(D')$, e.g.,
$$R\pi_*(\OO_Y): [\pi'_*(\rho^*L(D'))\to \pi'_*\rho^*L(D)|_{E'}],$$
where $\pi':Y\to B$ is the projection, $D'=\rho^{-1}(D)$, $E'=\rho^{-1}(E)$.
Note that $D'\to D$ and $E'\to E$ are isomorphisms, and the assertion follows from the commutativity of the squares
\begin{diagram}
\pi_*(L(D)|_E)&\rTo{\sim}&\pi_*(\rho_*\FF\ot L(D)|_E) \\
\dTo{}&&\dTo{}\\
\pi'_*(\rho^*L(D')|_{E'})&\rTo{\sim}&\pi'_*(\FF\ot\rho^*L(D')|_{E'})
\end{diagram}
\begin{diagram}
\pi_*(L(D)|_D)&\rTo{\sim}&\pi_*(\rho_*\FF\ot L(D)|_D) \\
\dTo{}&&\dTo{}\\
\pi'_*(\rho^*L(D')|_{D'})&\rTo{\sim}&\pi'_*(\FF\ot\rho^*L(D')|_{D'})
\end{diagram}
\end{proof}

\begin{lemma}\label{main-splitting-lem} The following diagram is commutative up to a sign
\begin{equation}\label{main-splitting-diagram}
\begin{diagram}
\BB(\om_{X/\ov{\SS}}^2)|_B&\rTo{\sim}& \BB(\om_{X/\ov{\SS}})^{-2}|_B\ot \BB(\OO_X)^{-1}(\De)|_B\\
\dTo{\sim}&&\dTo{\sim}\\
\BB(\om_{\wt{X}/B}^2(D_1+D_2))&\rTo{\sim}&\BB(\om_{\wt{X}/B})^{-2}\ot \BB(\OO_{\wt{X}})^{-1}\ot \Psi_1^{-1}\Psi_2^{-1}
\end{diagram}
\end{equation}
\end{lemma}

config: arxiv