elements of $\Omega_{X_{1}/S_{0}}\oplus\Omega_{X_{2}/S_{0}}$, the assertion
follows. ∎

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 $S_{0}=(t=0)\subset S$, we have a relative
Ramond node $q:S_{0}\to X_{0}=S_{0}\times SX$, so that the completion of
${\mathcal{O}}_{X}$ along $q(S_{0})$ is given by the standard generators
$z_{1},z_{2},\theta$ subject to $z_{1}z_{2}=t$, and with the distribution
generated by $\partial_{\theta}+\theta z_{i}\partial_{z_{i}}$ over $z_{i}\neq
0$.

######  Lemma 8.3.

(i) In the above situation the sheaf $j_{*}\Omega_{U/S}$ is freely generated
in a formal neighborhood of $q(S_{0})$, as an ${\mathcal{O}}_{X}$-module, by
global sections

$e=\frac{dz_{1}}{z_{1}}=-\frac{dz_{2}}{z_{2}},\ \ d\theta.$

(ii) We have an exact sequence on $X_{0}$,

$0\to{\mathcal{K}}\to\Omega_{X_{0}/S_{0}}\to
j_{*}\Omega_{U/S}|_{X_{0}}\to{\mathcal{C}}_{0}\to 0$

where ${\mathcal{K}}\simeq q_{*}{\mathcal{O}}_{S_{0}}$ is generated by
$z_{1}dz_{2}=-z_{2}dz_{1}$ and ${\mathcal{C}}_{0}\simeq
q_{*}{\mathcal{O}}_{S_{0}}$ is generated by the image of $e$. Furthermore, the
map of ${\mathcal{O}}_{S_{0}}$-modules
$\pi_{*}{\mathcal{K}}\to\pi_{*}\Omega_{X_{0}/S_{0}}$ is an embedding of a
direct summand.

###### Proof.

The proof is similar to that of Lemma 8.2 but is much easier since both
$\omega_{X/S}$ and $\omega_{X/S}^{2}$ are locally free near the Ramond node.
Namely, $\omega_{X/S}$ is freely generated by the section glued from
$\frac{1}{z_{1}}[dz_{1}|d\theta]$ and $-\frac{1}{z_{2}}[dz_{2}|d\theta]$, The
map $\kappa$ sends a generator of $\omega_{X/S}^{2}$ to $e-\theta d\theta$.
Since $\delta$ sends $d\theta$ to a generator of $\omega_{X/S}$, we deduce
that $e$ and $d\theta$ freely generate $j_{*}\Omega_{U/S}$ over
${\mathcal{O}}_{X}$.

The map $\Omega_{X_{0}/S_{0}}\to j_{*}\Omega_{U/S}|_{X_{0}}$ sends $dz_{1}$ to
$z_{1}e$, $dz_{2}$ to $-z_{2}e$, and $d\theta$ to $d\theta$. This easily
implies that its kernel is generated by $z_{2}dz_{1}=-z_{1}dz_{2}$, while its
cokernel is generate by the image of $e$. ∎

###  8.2. Boundary Cartier divisor

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

$[\Omega_{X/S}\to j_{*}\Omega_{U/S}]$

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
$\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to j_{*}\Omega_{U/S}]$ is well
defined. Similarly to the theory in the even case (see [20]), we can define a
canonical trivialization of this line bundle away from the locus of nodal
supercurves.

######  Proposition 8.4.

The canonical trivialization $c$ of
${\mathcal{B}}:=\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to j_{*}\Omega_{U/S}]$
away from the nodal locus extends to a regular section of ${\mathcal{B}}$ 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
${\mathcal{B}}$ and some local function $t$ whose reduction modulo nilpotents
gives an equation of the corresponding reduced divisor.

###### 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
$\pi_{*}[\Omega_{X/S}\to j_{*}\Omega_{U/S}]$ will be represented by the compex
of supervector bundles

$\pi_{*}(\Omega_{X/S}(D))\rTo{\iota}\pi_{*}(j_{*}\Omega_{U/S}(D)).$

To understand this map, we first restrict it to $X_{0}$. We have an exact
sequence on $S$,

$0\to\pi_{*}{\mathcal{K}}\to\pi_{*}(\Omega_{X_{0}/S_{0}}(D))\to\pi_{*}(j_{*}\Omega_{U/S}(D)|_{S_{0}})\to\pi_{*}{\mathcal{C}}_{0}\to
0,$

where both $\pi_{*}{\mathcal{K}}$ and $\pi_{*}{\mathcal{C}}_{0}$ are locally
free of rank $2|2$. Furthermore, $\pi_{*}{\mathcal{K}}$ embeds into
$\pi_{*}(\Omega_{X_{0}/S_{0}}(D))$ as a subbundle, and we have an
${\mathcal{O}}_{S_{0}}$-basis (8.2) of $\pi_{*}{\mathcal{K}}$. 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
$\pi_{*}\Omega_{X_{0}/S_{0}}(D)$.

We can choose liftings of the ${\mathcal{O}}_{S_{0}}$-basis (8.2) in
$\pi_{*}{\mathcal{K}}$ to ${\mathcal{O}}_{S}$-independent sections
$b_{1},b_{2},b_{3},b_{4}$ in $\pi_{*}(\Omega_{X/S}(D))$, so that

$b_{1}\equiv z_{1}dz_{2}\mod(t),\ b_{2}\equiv\theta_{2}dz_{1}\mod(t),\
b_{3}\equiv\theta_{1}dz_{2}\mod(t),\ b_{4}\equiv\theta_{2}d\theta_{1}\mod(t).$

Then we can extend it to a basis $(b_{1},\ldots,b_{n})$ of
$\pi_{*}(\Omega_{X/S}(D))$ so that the elements
$(\iota(b_{5}),\ldots,\iota(b_{n}))$ form a basis of a free
${\mathcal{O}}_{S}$-submodule in $\pi_{*}(j_{*}\Omega_{U/S}(D))$, which
projects modulo $(t)$ to a basis of the image of
$\pi_{*}\Omega_{X_{0}/S_{0}}(D)$.

Next, we adjust our choice of $b_{1}$ so that

$b_{1}\equiv z_{1}dz_{2}\mod(t^{2}).$

Here the right-hand side can be viewed as a section of

$\pi_{*}(\Omega_{X/S}(D))/(t^{2})=\pi_{*}(\Omega_{X_{0}^{(1)}/S_{0}^{(1)}}(D)),$

where we consider the base change $X_{0}^{(1)}\to S_{0}^{(1)}$, where
$S_{0}^{(1)}\subset S$ is given by the ideal $t^{2}$. Namely, we observe that
near the node one has $I_{q}^{2}\subset(z_{1},z_{2})$, which implies that
$z_{1}dz_{2}=-z_{2}dz_{1}$ is killed by $I_{q}^{2}$ on $X_{0}^{(1)}$ (due to
the relation $z_{1}z_{2}=-t^{2}$). Thus, we can lift $z_{1}dz_{2}\mod(t^{2})$
to a section $b_{1}\in\pi_{*}(\Omega_{X/S}(D))$.

Now let us analyse the images of $b_{1},\ldots,b_{4}$ under $\iota$. We have
$\iota(b_{i})=tc_{i}$ for some $c_{i}\in\pi_{*}(j_{*}\Omega_{U/S}(D))$, for
$i=2,3,4$. In the case of $b_{1}$ we know that $\iota(b_{1})=t^{2}c_{1}$ for
some $c_{1}\in\pi_{*}(j_{*}\Omega_{U/S}(D))$.

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 $z_{1}dz_{2}+t^{2}x$, it maps to

$-z_{1}z_{2}e+t^{2}x=t^{2}(e+x),$

where $x$ comes from $\Omega_{\hat{X}/S}$. Hence,

$c_{1}\equiv e\mod\Omega_{\hat{X}/S}.$

Similarly, we compute

$\theta_{2}dz_{1}=\theta_{2}z_{1}e=t\theta_{1}e,$
$\theta_{1}dz_{2}=-\theta_{1}z_{2}e=t\theta_{2}e,$ $\theta_{2}d\theta_{1}=tf.$

Hence, we get

$c_{2}\equiv\theta_{1}e\mod\Omega_{\hat{X}/S},\ \
c_{3}\equiv\theta_{2}e\mod\Omega_{\hat{X}/S},\ \ c_{4}\equiv
f\mod\Omega_{\hat{X}/S}.$

It follows that $c_{1},c_{2},c_{3},c_{4}$ project to an
${\mathcal{O}}_{S}$-basis of $\pi_{*}{\mathcal{C}}_{0}$. Hence,
$c_{1},c_{2},c_{3},c_{4}$ are linearly indpendent over ${\mathcal{O}}_{S}$,
and can be extended to an ${\mathcal{O}}_{S}$-basis $(c_{1},\ldots,c_{n})$ of
$\pi_{*}(j_{*}\Omega_{U/S}(D))$ 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 8.3. In
this case we can choose bases $(b_{1},\ldots,b_{n})$ of
$\pi_{*}(\Omega_{X/S}(D))$ and $(c_{1},\ldots,c_{n})$ of
$\pi_{*}(j_{*}\Omega_{U/S}(D))$, such that $b_{1}\equiv
z_{1}dz_{2}\mod(t^{2})$, $\iota(b_{1})=-tc_{1}$ (since
$z_{1}dz_{2}=-z_{1}z_{2}e=-te$ in $j_{*}\Omega_{U/S}(D)$), and
$\iota(b_{i})=c_{i}$ for $i>1$. This shows that the Berezinian is $t$. ∎

The above Proposition gives a natural definition of the boundary divisor in
the moduli space $\overline{\SS}$ as an effective Cartier divisor. More
precisely, let $k:\SS^{\prime}\hookrightarrow\overline{\SS}$ 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 $\overline{\SS}$ is smooth, for
any vector bundle ${\mathcal{V}}$ over $\overline{\SS}$, the natural map
${\mathcal{V}}\to k_{*}k^{*}{\mathcal{V}}$ is an isomorphism. Let us consider
the line bundle

${\mathcal{L}}:=\operatorname{Ber}\pi_{*}[\Omega_{X/\overline{\SS}}\to
j_{*}\Omega_{U/\overline{\SS}}]$

over $\overline{\SS}$. By Proposition 8.4, the canonical trivialization $c$ of
${\mathcal{L}}$ over the smooth locus gives a regular global section of
$k^{*}{\mathcal{L}}$ and hence of ${\mathcal{L}}\simeq
k_{*}k^{*}{\mathcal{L}}$. We define the effective Cartier divisor $\Delta$ to
be the vanishing divisor of this global section. Note that by definition, we
have an isomorphism

${\mathcal{O}}(\Delta)\simeq{\mathcal{L}}\simeq\operatorname{Ber}R\pi_{*}(j_{*}\Omega_{U/S})\otimes\operatorname{Ber}^{-1}R\pi_{*}(\Omega_{X/S}).$
(8.4)

The proof of Proposition 8.4 also yields the following characterization of the
divisor $\Delta$.

######  Lemma 8.5.

The effective Cartier divisor $\Delta$ is a unique Cartier divisor supported
on the locus of nodal supercurves with the following two properties:

  * •

for any stable supercurve $X_{0}$ with a single NS node $q$ there exists an
étale neighborhood $S$ of the corresponding point $[X_{0}]$ in the moduli
space and an étale neighborhood $X_{S}$ of $q$ in the family $X\to S$ induced
by the universal family such that ${\mathcal{O}}_{X_{S}}$ is generated over
${\mathcal{O}}_{S}$ by $z_{1},z_{2},\theta_{1},\theta_{2}$ subject to
relations (7.10), where $t$ is a local equation of $\Delta$ on $S$;

  * •

for any stable supercurve $X_{0}$ with a single Ramond node $q$ there exists
an étale neighborhood $S$ of $[X_{0}]$ in the moduli space and an étale
neighborhood $X_{S}$ of $q$ in the family $X\to S$ induced by the universal
family such that ${\mathcal{O}}_{X_{S}}$ is generated over ${\mathcal{O}}_{S}$
by $z_{1},z_{2},\theta$ subject to the relation $z_{1}z_{2}=t$, where $t$ is a
local equation of $\Delta$ on $S$;

###### Proof.

These conditions clearly characterize $\Delta\cap\SS^{\prime}$. To show
uniqueness of an extension to $\overline{\SS}$, we observe that if
${\mathcal{I}}\subset{\mathcal{O}}_{\overline{\SS}}$ is an invertible ideal
sheaf then ${\mathcal{I}}$ is identified with $k_{*}k^{*}{\mathcal{I}}\subset
k_{*}{\mathcal{O}}_{\SS^{\prime}}={\mathcal{O}}_{\overline{\SS}}$. ∎

###  8.3. Another definition of the boundary divisor and the normal crossing
property

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

######  Lemma 8.6.

The natural morphism
${\mathcal{A}}_{X}\to\pi^{*}{\mathcal{T}}_{\overline{\SS}}$ of sheaves on $X$
factors through a morphism
${\mathcal{A}}_{X}\to\pi^{-1}{\mathcal{T}}_{\overline{\SS}}$.

###### Proof.

This follows the fact that ${\mathcal{A}}_{X}=j_{*}{\mathcal{A}}_{U}$, where
$U\subset X$ is the complement to the nodes (see the proof of Theorem 5.4),
and from the corresponding statement for the smooth locus $U\to\overline{\SS}$
(see Eq. (7.8)). ∎

Let us consider the natural morphism

$\pi_{*}({\mathcal{A}}_{X}/{\mathcal{A}}_{X/\overline{\SS}})\to{\mathcal{T}}_{\overline{\SS}}$
(8.5)

induced by the map
${\mathcal{A}}_{X}\to\pi^{-1}{\mathcal{T}}_{\overline{\SS}}$.

######  Proposition 8.7.

The ${\mathcal{O}}_{\overline{\SS}}$-module
$\pi_{*}({\mathcal{A}}_{X}/{\mathcal{A}}_{X/\overline{\SS}})$ is locally free
of rank $(3g-3|2g-2)$ over $\overline{\SS}$, and the Cartier divisor
associated with the Berezinian of the map (8.5) coincides with $\Delta$.
Furthermore, $\Delta$ is a normal crossing divisor.

###### Proof.

Note that (8.5) is an isomorphism over the locus of smooth supercurves. By
Lemma 7.5, in an étale neighborhood $S$ of a point $[X_{0}]$, where $X_{0}$
has a single NS node, the image of the morphism
${\mathcal{A}}_{X}\to\pi^{-1}{\mathcal{T}}_{S}$ is
$\pi^{-1}{\mathcal{T}}_{S,S_{0}}$, where $S_{0}$ is the divisor $(t)$. Hence,
in this case (8.5) is the embedding
${\mathcal{T}}_{B,B_{0}}\hookrightarrow{\mathcal{T}}_{B}$, and the Berezinian
of this morphism is $t$. By Lemma 7.8, the similar statement holds in a
neighborhood of a curve with a single Ramond node. This implies that
$\pi_{*}({\mathcal{A}}_{X}/{\mathcal{A}}_{X/\overline{\SS}})$ is locally free
over $\SS^{\prime}\subset\overline{\SS}$, and the divisor of the Berezinian of
(8.5) over $\SS^{\prime}$ is $\Delta\cap\SS^{\prime}$.

Nex, let us study the situation near a point $[X_{0}]$ of $\overline{\SS}$,
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 étale neighborhood
$B$ of $[X_{0}]$, together with a smooth map
$(t_{1},\ldots,t_{k}):B\to{\mathbb{A}}^{k}$ 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 ${\mathcal{T}}_{B,D}\subset{\mathcal{T}}_{B}$ be the subsheaf of
derivations preserving $D$. Note that
${\mathcal{T}}_{B,D}=\cap{\mathcal{T}}_{B,D_{i}}$ where $D_{i}$ is the divisor
$t_{i}=0$.

Let us consider the restriction $\pi:X\to B$ of the universal family and set

$\overline{{\mathcal{A}}}_{X}:={\mathcal{A}}_{X}/{\mathcal{A}}_{X/B}\subset\pi^{*}{\mathcal{T}}_{B}.$

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

$\overline{{\mathcal{A}}}_{X}|_{U_{i}}=\pi_{i}^{-1}{\mathcal{T}}_{B,D_{i}}.$

Thus, for each $i=1,\ldots,k$, we have an inclusion

$\pi_{*}(\overline{{\mathcal{A}}}_{X})\subset\pi_{i*}(\overline{{\mathcal{A}}}_{X}|_{U_{i}})={\mathcal{T}}_{B,D_{i}}\subset{\mathcal{T}}_{B}.$

Hence, we deduce the inclusion

$\pi_{*}(\overline{{\mathcal{A}}}_{X})\subset{\mathcal{T}}_{B,D}.$

On the other hand, we claim that there is an inclusion

$\pi^{-1}{\mathcal{T}}_{B,D}\subset\overline{{\mathcal{A}}}_{X}.$

Indeed, it is enough to check this over each $U_{i}$. But then we have

$\pi^{-1}{\mathcal{T}}_{B,D}|_{U_{i}}\subset\pi_{i}^{-1}{\mathcal{T}}_{B,D_{i}}=\overline{{\mathcal{A}}}_{X},$

as required. Hence, passing to $\pi_{*}(?)$, we derive the inclusion

${\mathcal{T}}_{B,D}=\pi_{*}\pi^{-1}{\mathcal{T}}_{B,D}\subset\pi_{*}\overline{{\mathcal{A}}}_{X}.$

as claimed.

Thus, we get $\pi_{*}\overline{{\mathcal{A}}}_{X}={\mathcal{T}}_{B,D}$. Hence,
$\pi_{*}\overline{{\mathcal{A}}}_{X}$ is locally free and the Berezinian of
the morphism ${\mathcal{T}}_{B,D}\to{\mathcal{T}}_{B}$ has the required form.
∎

###  8.4. NS and Ramond boundary components as effective Cartier divisors

Note that $\Delta\cap\SS^{\prime}$ 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
$\SS^{\prime}$ to effective Cartier divisors $\Delta_{NS}$ and $\Delta_{R}$ on
$\overline{\SS}$ such that $\Delta=\Delta_{NS}+\Delta_{R}$.

First, we are going to define the divisor $\Delta_{NS}$ giving the NS
component. For this let us consider the structure map

$\delta:\Omega_{X/\overline{\SS}}\to\omega_{X/\overline{\SS}}(R).$

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
$\overline{\SS}$,

${\mathcal{I}}_{\delta}:=\operatorname{Ann}\pi_{*}\operatorname{coker}(\delta),$

supported on the locus of stable supercurves with at least one NS node.

######  Proposition 8.8.

(i) The ideal sheaf ${\mathcal{I}}_{\delta}$ defines an effective Cartier
divisor $\Delta_{NS}$ on $\overline{\SS}$, which coincides with $\Delta$ in a
neighborhood of any point $[X_{0}]$ corresponding to a stable supercurve
$X_{0}$ with only NS nodes.

(ii) There exists a unique effective Cartier divisor $\Delta_{R}$ supported on
the locus of stable supercurves with at least one Ramond node, such that

$\Delta=\Delta_{NS}+\Delta_{R}.$

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 étale neighborhood
$B$ of $[X_{0}]$ in $\overline{\SS}$ with a smooth morphism
$t_{1},\ldots,t_{k}:B\to{\mathbb{A}}^{k}$ such that $t_{1}\ldots t_{r}$ is an
equation of $\Delta_{NS}$ and $t_{r+1}\ldots t_{k}$ is an equation of
$\Delta_{R}$.

###### 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 étale neighborhood $B$ of
$[X_{0}]$, equipped with a smooth map
$(t_{1},\ldots,t_{k}):B\to{\mathbb{A}}^{k}$, such that $t_{i}$ gives the
universal deformation of the node $q_{i}$. Then the ideal of $\Delta$ is
generated by $t_{1}\ldots t_{k}$.

On the other hand, using the description of $\omega_{X/S}$ by generators and
relations (see Lemma 7.4), we see that in the formal neighborhood of each NS
node $q_{i}$, $\operatorname{coker}(\delta)$ is generated by $s_{0}$ subject
to the relations

$\theta_{1}s_{0}=\theta_{2}s_{0}=z_{1}s_{0}=z_{2}s_{0}=t_{i}s_{0}=0.$

It is easy to deduce from this that the annihilator of
$\pi_{*}\operatorname{coker}(\delta)$ is generated by $t_{1}\ldots t_{r}$.
This implies all the assertions. ∎

###  8.5. The NS node boundary components

We will use the following construction of gluing two superschemes along a
closed subscheme.

Assume that $X$, $X^{\prime}$ and $Y$ are superschemes, $i:Y\to X$ and $i:Y\to
X^{\prime}$ are closed embeddings. We want to define a new superscheme $Z$ by
gluing $X$ and $X^{\prime}$ along $Y$. As a topological space we can define
$Z$ to the usual gluing of the topological spaces of $X$ and $X^{\prime}$
along $Y$, so that we have closed embeddings $j:X\to Z$,
$j^{\prime}:X^{\prime}\to Z$, so that $k:=j_{1}\circ i=j_{2}\circ i$. Hence,
we have two homomorphisms of sheaves of rings

$\phi:j_{*}{\mathcal{O}}_{X}\to k_{*}{\mathcal{O}}_{Y},\ \
\phi^{\prime}:j^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}\to
k_{*}{\mathcal{O}}_{Y},$

and we define ${\mathcal{O}}_{X}$ to be the subsheaf of
$j_{*}{\mathcal{O}}_{X}\oplus j^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}$,
namely, the preimage of the diagonal $k_{*}{\mathcal{O}}_{Y}\subset
k_{*}{\mathcal{O}}_{Y}\oplus k_{*}{\mathcal{O}}_{Y}$ under
$\phi\oplus\phi^{\prime}$.

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

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

######  Lemma 8.9.

The glued superscheme $Z/S$ has a natural stable supercurve structure such
that the derivation $\delta$ on $Z$ is defined as the composition

${\mathcal{O}}_{Z}\to j_{*}{\mathcal{O}}_{X}\oplus
j^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}\rTo{(j_{*}\delta,j^{\prime}_{*}\delta^{\prime})}j_{*}\omega_{X/S}\oplus
j^{\prime}_{*}\omega_{X^{\prime}/S}\to\omega_{Z/S}.$

###### Proof.

If $U\subset X$, $U^{\prime}\subset X^{\prime}$ are smooth loci, then
$(U\setminus P)\sqcup(U^{\prime}\setminus P^{\prime})$ 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, $\delta^{-}$ induces an isomorphism of
${\mathcal{O}}_{Z}^{-}$ with $\omega_{Z}^{-}$. Since the base is even, we have
identifications

${\mathcal{O}}_{X}={\mathcal{O}}_{C}\oplus L,\ \
{\mathcal{O}}_{X^{\prime}}={\mathcal{O}}_{C^{\prime}}\oplus L^{\prime},$

and the smooth marked points $P\subset C$, $P^{\prime}\subset C^{\prime}$, so
that the embedding of $P$ into $X$ corresponds to the projection

${\mathcal{O}}_{X}\to{\mathcal{O}}_{C}\to{\mathcal{O}}_{P}.$

Note that $L$ and $L^{\prime}$ are generalized spin-structures, i.e., we have
an isomorphism $L\rTo{\sim}\underline{\operatorname{Hom}}(L,\omega_{C})$
(which corresponds to $\delta^{-}$ on $X$), and similarly, for $L^{\prime}$.
Furthermore, we know that $L$ (resp., $L^{\prime}$) is locally free near $P$
(resp., $P^{\prime}$).

The glued superscheme $Z$ has the underlying nodal curve $Z_{0}$, which is
glued from $C$ and $C^{\prime}$ along $P\simeq P^{\prime}$, and

${\mathcal{O}}_{Z}\simeq{\mathcal{O}}_{Z_{0}}\oplus j_{*}L\oplus
j^{\prime}_{*}L^{\prime}.$

It is well known that the natural map

$j_{*}L\oplus
j^{\prime}_{*}L^{\prime}\to\underline{\operatorname{Hom}}(j_{*}L\oplus
j^{\prime}_{*}L^{\prime},\omega_{Z_{0}})$

is an isomorphism, i.e., $j_{*}L\oplus j^{\prime}_{*}L^{\prime}$ is a
generalized spin-structure. It is easy to see that above map is precisely
$\delta^{-}$, so this proves that $Z/S$ is a stable supercurve. ∎

Let $\overline{\SS}_{g;m,n}$ 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
$\overline{\SS}_{g_{1};m_{1}+1,n_{1}}\times\overline{\SS}_{g_{2};m_{2}+1,n_{2}}$,
pulled back from each factor and using the last NS puncture on each of them.
This leads to a morphism

$\overline{\SS}_{g_{1};m_{1}+1,n_{1}}\times\overline{\SS}_{g_{2};m_{2}+1,n_{2}}\to\overline{\SS}_{g_{1}+g_{2};m_{1}+m_{2},n_{1}+n_{2}}.$
(8.6)

Similarly, if $X/S$ is a stable supercurve with two disjoint NS-punctures
$P,P^{\prime}\subset X$, then we can glue $P$ with $P^{\prime}$ and get a new
stable supercurve with a non-separating node. This leads to a morphism

$\overline{\SS}_{g;m+2,n}\to\overline{\SS}_{g+1;m,n}$ (8.7)

######  Lemma 8.10.

Both morphisms (8.6) and (8.7) induce embeddings of codimension $1|0$ on
tangent spaces and factor through the divisor $\Delta_{NS}$. Furthermore, near
a stable supercurve with a single separating (resp., non-separating) NS node,
the divisor $\Delta_{NS}$ coincides with the schematic image of (8.6) (resp.,
(8.7)).

###### Proof.

The idea is to use exact sequences (4.3). Let $X$ be a stable supercurve with
punctures and a fixed NS-node $q\in X$, and let $\rho:\widetilde{X}\to X$ be
the normalization at $q$, equipped with the two NS punctures over $q$. Then by
Proposition 3.6(i), we have

${\mathcal{A}}_{X}\simeq\rho_{*}{\mathcal{A}}_{\widetilde{X}}$

(where we take into account all the punctures on both sides). Hence, the
natural map

$H^{1}(\widetilde{X},{\mathcal{A}}_{\widetilde{X}})\to
H^{1}(X,{\mathcal{A}}_{X})$

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 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 $\Delta_{NS}$, we have to check
that the pull-back of the morphism
${\mathcal{O}}\to{\mathcal{O}}(\Delta_{NS})$ 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 $\Delta_{NS}$ 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 (8.6) and (8.7) 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 $\Delta_{NS}$ is the schematic image of the morphism. ∎

###  8.6. The Ramond node boundary components

Gluing along two Ramond punctures is more subtle.

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

######  Lemma 8.11.

The map

$\delta|_{R}:\Omega_{X/S}|_{R}\to\omega_{X/R}(R)|_{R}\simeq\omega_{R/S}$

factors through the canonical projection $\Omega_{X/S}|_{R}\to\Omega_{R/S}$,
and induces an isomorphism

$\overline{\delta}_{R}:\Omega_{R/S}\to\omega_{R/S}.$

This induces a trivialization of $(\Omega_{R/S})^{\otimes 2}$ (or
equivalently, of $\omega_{R/S}^{\otimes 2}$) which is locally given by
$(d\theta)^{2}$, for $\theta$ such that
${\mathcal{O}}_{R}={\mathcal{O}}_{S}[\theta]$,

###### Proof.

Recall that in appropriate étale local coordinates $(z,\theta)$, $R$ is given
by $(z=0)$ and we have

$\delta(dz)=z\theta[\frac{dz}{z}|d\theta],\ \
\delta(d\theta)=[\frac{dz}{z}|d\theta].$

It follows that $\delta|_{R}(dz)=0$, and the induced map
$\overline{\delta}_{R}$ is given by

$\overline{\delta}_{R}(d\theta)=\overline{b},$

where $\overline{b}$ is the generator of $\omega_{R/S}$ corresponding to
$[\frac{dz}{z}|d\theta]$.

Since the relative dimension of $R$ is $0|1$, we have in fact an isomorphism
$\omega_{R/S}^{-1}\simeq\Omega_{R/S}$. Thus, we can view
$\overline{\delta}_{R}$ as a trivializing section of $(\Omega_{R/S})^{\otimes
2}$. ∎

The above Lemma implies that every Ramond puncture $R$ has a preferred system
of étale local relative coordinates $\theta_{i}$, such that over intersections
one has $\theta_{j}=\pm\theta_{i}+a_{ij}$, where $a_{ij}$ are functions on the
base. Namely, we require that $(d\theta_{i})^{2}$ is the canonical
trivialization of $(\Omega_{R/S})^{\otimes 2}$. In other words, we have a
canonical principal bundle ${\mathcal{P}}_{R}\to S$ with the structure
supergroup $\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$.

We can restate the above structure in more invariant terms. Let $\pi:R\to S$
denote the projection. First, we observe that

$\Phi_{R}:=\pi_{*}{\mathcal{O}}_{R}/{\mathcal{O}}_{S}\simeq\operatorname{Ber}^{-1}\pi_{*}{\mathcal{O}}_{R}$
(8.8)

is a line bundle of rank $0|1$ on $S$. Furthermore, we have a canonical
isomorphism of odd line bundles on $R$,

$\pi^{*}\Phi_{R}\rTo{\sim}\Omega_{R}$

induced by the de Rham differential
${\mathcal{O}}_{S}/{\mathcal{O}}_{R}\to\Omega_{R}$. Furthermore, the
trivialization of $(\Omega_{R})^{\otimes 2}$ in Lemma 8.11 actually comes from
the canonical trivialization of $\Phi_{R}^{2}$ on $S$.

Let $\Theta\subset\operatorname{tot}(\Phi_{R})$ be the
$\operatorname{\mathbb{Z}}/2$-torsor over $S$ corresponding to $\Phi_{R}$.
Then the principal bundle ${\mathcal{P}}_{R}\to S$ can be identified with the
preimage of $\Theta$ in $\operatorname{tot}(\pi_{*}{\mathcal{O}}_{R})$ under
the natural projection
$p:\pi_{*}{\mathcal{O}}_{R}\to\pi_{*}{\mathcal{O}}_{R}/{\mathcal{O}}_{S}$.

######  Lemma 8.12.

The group scheme $\operatorname{Aut}(\Phi_{R})\to S$ can be identified with
the group of automorphisms of $\pi_{*}{\mathcal{O}}_{R}$, which are identity
on ${\mathcal{O}}_{S}$, and induce $\pm\operatorname{id}$ on
$\pi_{*}{\mathcal{O}}_{R}/{\mathcal{O}}_{S}$. This group scheme is an
extension of $\operatorname{\mathbb{Z}}/2$ by the line bundle $\Phi_{R}^{-1}$.

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

######  Lemma 8.13.

Assume that the isomorphism $\alpha:R\rTo{\sim}R^{\prime}$ is such that the
induced isomorphism

$\alpha^{*}\omega_{R^{\prime}/S}^{\otimes 2}\rTo{\sim}\omega_{R/S}^{\otimes
2}$

is equal to $-1$, where we use the trivializations of $\omega_{R/S}^{\otimes
2}$ and $\omega_{R^{\prime}/S}^{\otimes 2}$ coming from the supercurve
structure on $X$. Then the glued superscheme $Z/S$ has a natural stable
supercurve structure, such that the derivation $\delta$ is uniquely determined
from the commutative diagram

###### Proof.

To check that $\delta$ is well defined, we need to check that the image of the
composition $j_{*}\delta\circ r$ belongs to $\omega_{Z/S}$. For this we can
argue locally. Let $(z,\theta)$ be the standard local coordinates on $X$ near
the puncture $R$, so that $R$ is given by $(z=0)$, and

$\delta(dz)=z\theta[\frac{dz}{z}|d\theta],\ \
\delta(d\theta)=[\frac{dz}{z}|d\theta].$

Let also $(z^{\prime},\theta^{\prime})$ be similar coordinates near
$R^{\prime}$. Then $\alpha^{*}(d\theta^{\prime})^{2}=-d\theta$, so changing
$\theta$ to $\pm\theta+a$, we can assume that
$i\alpha^{*}(\theta^{\prime})=\theta$. Let us set
$\overline{\theta}=i\theta^{\prime}$. Then we have

$\delta(dz^{\prime})=z^{\prime}\overline{\theta}[\frac{dz}{z}|d\overline{\theta}],\
\
\delta(d\overline{\theta})=-[\frac{dz^{\prime}}{z^{\prime}}|d\overline{\theta}].$

Then we have relative coordinates on the glued scheme $Z$,
$(z,z^{\prime},\theta)$, where $zz^{\prime}=0$, $\theta$ restricts to $\theta$
near $R$ and to $\overline{\theta}$ near $R^{\prime}$. Since $\delta(z)$ has
no pole at $R$, we see that $(\delta(z),0)$ belongs to $\omega_{Z/S}$.
Similarly, $(0,\delta^{\prime}(z^{\prime}))$ belongs to $\omega_{Z/S}$.
Finally,

$(\delta(\theta),\delta^{\prime}(\overline{\theta}))=([\frac{dz}{z}|d\theta],-[\frac{dz^{\prime}}{z^{\prime}}|d\overline{\theta}]),$

which comes from a section of $\omega_{Z/S}$.

Over the point, our construction can be recast as follows. We start with a
Ramond spin curve $(C,p,p^{\prime},L)$, together with an identification

$\alpha:L|_{p}\simeq L|_{p^{\prime}},$

such that $\alpha^{2}=-1$ (where we use trivializations of
$\omega_{C}(p)|_{p}$ and $\omega_{C}(p^{\prime})|_{p^{\prime}}$). We glue $p$
with $p^{\prime}$ and get a nodal curve $Z_{0}$, and then descend $L$ to a
line bundle ${\mathcal{L}}$ over $Z_{0}$ using $\alpha$. Then we have a
natural isomorphism ${\mathcal{L}}^{\otimes 2}\simeq\omega_{Z_{0}}$, and one
can easily check that the corresponding isomorphism
${\mathcal{L}}\rTo{\sim}\underline{\operatorname{Hom}}({\mathcal{L}},\omega_{Z_{0}})$
is induced by $\delta^{-}$, where $\delta$ is defined as above. ∎

The choice of an isomorphism $\alpha$ can be interpreted in terms of the
principal $\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$-bundles
${\mathcal{P}}_{R}$ and ${\mathcal{P}}_{R^{\prime}}$ as follows. For every
$c\in\operatorname{\mathbb{C}}^{*}$ let us denote by $[c]$ the automorphism of
$\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$ given by the rescaling
by $c$ on ${\mathbb{A}}^{0|1}$ (and trivial on $\operatorname{\mathbb{Z}}/2$).
Then a choice of $\alpha$ is equivalent to a choice of an isomorphism of
$\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$-bundles,

${\mathcal{P}}_{R}\to[i]_{*}{\mathcal{P}}_{R^{\prime}},$

where $[i]_{*}{\mathcal{P}}_{R^{\prime}}$ is the push-out of
${\mathcal{P}}_{R^{\prime}}$ 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
$\overline{\SS}_{g;m,n+2}$. Let
${\mathcal{P}}_{n+1}\to\overline{\SS}_{g;m,n+2}$ and
${\mathcal{P}}_{n+2}\to\overline{\SS}_{g;m,n+2}$ be the corresponding
principal $\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$-bundles. Let
$\operatorname{Isom}({\mathcal{P}}_{n+1},[i]_{*}{\mathcal{P}}_{n+2})\to\overline{\SS}_{g;m,n+2}$
denote the bundle of isomorphisms between these
$\operatorname{\mathbb{Z}}/2\ltimes{\mathbb{A}}^{0|1}$-bundles. Note that it
is also a principal bundle with the group
$\operatorname{\mathbb{Z}}/2\times{\mathbb{A}}^{0|1}$.

Then the above gluing construction gives a family of stable supercurves with a
Ramond node over
$\operatorname{Isom}({\mathcal{P}}_{n+1},[i]_{*}{\mathcal{P}}_{n+2})$, so we
get a morphism

$\operatorname{Isom}({\mathcal{P}}_{n+1},[i]_{*}{\mathcal{P}}_{n+2})\to\overline{\SS}_{g+1;m,n}$
(8.9)

Similarly, we can glue the last Ramond punctures $R,R^{\prime}$ on the stable
supercurves $X,X^{\prime}$ over the moduli space
$\overline{\SS}_{g_{1};m_{1},n_{1}+1}\times\overline{\SS}_{g_{2};m_{2},n_{2}+1}$
pulled back from each factor. This gives a morphism

$\operatorname{Isom}({\mathcal{P}}_{R},[i]_{*}{\mathcal{P}}_{R^{\prime}})\to\overline{\SS}_{g_{1}+g_{2};m_{1}+m_{2},n_{1}+n_{2}}.$
(8.10)

Note that in this case $n_{1}$ and $n_{2}$ are odd, so $n_{1}+n_{2}\geq 2$.

######  Lemma 8.14.

Both morphisms (8.9) and (8.10) induce embeddings of codimension $1|0$ on
tangent spaces and factor through the divisor $\Delta_{R}$. Furthermore, near
a stable supercurve with a single non-separating Ramond node the divisor
$\Delta_{R}$ coincides with the schematic image of (8.9).

###### Proof.

Let $X$ be the stable supercurve obtained by gluing two Ramond punctures
$R,R^{\prime}$ on a stable supercurve $\widetilde{X}$, via an isomorphism
$\alpha:R\to R^{\prime}$ as in Lemma 8.13. Note that we have an exact sequence

$0\to\Pi\operatorname{\mathbb{C}}\to
T_{\operatorname{Def}(\widetilde{X},\alpha)}\to
T_{\operatorname{Def}(\widetilde{X})}\to 0,$

where the odd line $\Pi\operatorname{\mathbb{C}}$ corresponds to infinitesimal
deformations of $\alpha$. By Proposition 3.6(i), we have

${\mathcal{A}}_{X}^{+}\simeq\rho_{*}{\mathcal{A}}_{\widetilde{X}}^{+}.$

As in Lemma 8.10 we deduce from this that

$T^{+}_{\operatorname{Def}(\widetilde{X},\alpha)}\to
T^{+}_{\operatorname{Def}(X)}$

is an embedding of codimension $1$.

Next, we claim that the map between the spaces of odd infinitesimal
deformations,

$T^{-}_{\operatorname{Def}(\widetilde{X},\alpha)}\to
T^{-}_{\operatorname{Def}(X)}$

is an isomorphism. Indeed, we know that every odd deformation of $X$ is
locally trivial, i.e.,

$T^{-}_{\operatorname{Def}(X)}\simeq H^{1}(X,{\mathcal{A}}_{X}^{-}).$

Also, by Proposition 3.6(i), we have an exact sequence

$0\to\operatorname{\mathbb{C}}\to H^{1}(X,{\mathcal{A}}_{X}^{-})\to
H^{1}(\widetilde{X},{\mathcal{A}}_{\widetilde{X}}^{-})\to 0$

so

$\dim T^{-}_{\operatorname{Def}(X)}=\dim
H^{1}(\widetilde{X},{\mathcal{A}}_{\widetilde{X}}^{-})+1=\dim
T^{-}_{\operatorname{Def}(\widetilde{X},\alpha)}.$

It remains to observe that every locally trivial deformation of $X$ can be
lifted to a deformation of $(\widetilde{X},\alpha)$. ∎

##  9\. Canonical line bundle on the moduli space of stable supercurves

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

###  9.1. What Kodaira-Spencer isomorphism gives for the canonical bundle

Set $\overline{\SS}=\overline{\SS}_{g,n_{NS},n_{R}}$. We are interested in the
canonical line bundle
$K_{\overline{\SS}}=\operatorname{Ber}^{-1}({\mathcal{T}}_{\overline{\SS}})$.
Over the smooth locus $\SS\subset\overline{\SS}$, from the Kodaira-Spencer
isomorphism we get an isomorphism (see Sec. 7.4)

$\operatorname{ber}(KS_{\overline{\SS}}):K_{\SS}\rTo{\sim}\operatorname{Ber}^{-1}(R^{1}\pi_{*}{\mathcal{L}}(X,P_{\bullet},R_{\bullet}))\simeq\operatorname{Ber}(R\pi_{*}{\mathcal{L}}(X,P_{\bullet},R_{\bullet})),$
(9.1)

where

${\mathcal{L}}(X,P_{\bullet},R_{\bullet}):=\omega_{X/S}^{-2}(-\sum_{i\in
I}D_{i}-2\sum_{j\in J}R_{j}).$

Recall that here $\omega_{X/S}^{-2}$ is the line bundle defined in Theorem
5.1, and $D_{i}$ are the divisors on the universal curve $X$ associated with
the NS punctures.

Our study in Sec. 7 of the behavior of the Kodaira-Spencer map in degenerating
families of supercurves leads to the following identification of
$K_{\overline{\SS}}$.

######  Proposition 9.1.

One has a natural isomorphism

$\operatorname{ber}(KS_{\overline{\SS}})^{-1}:K_{\overline{\SS}}\rTo{\sim}\operatorname{Ber}(R\pi_{*}{\mathcal{L}}(X,P_{\bullet},R_{\bullet}))(-\Delta),$

where $\Delta$ is the boundary divisor.

###### Proof.

We just have to check that isomorphism (9.1) acquires simple poles at all
generic points of $\Delta$. But this follows from the results of Sec. 7 and
our local description of $\Delta$ at points corresponding to stable
supercurves with single nodes. ∎

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

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

###  9.2. Isomorphisms for Berezinian bundles

Let $\pi:X\to S$ be a smooth proper morphism of relative dimension $1|1$. Then
we can define Deligne’s symbol $\langle L_{1}$, $L_{2}\rangle$ 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

$\langle{\mathcal{O}}_{X}(D),L\rangle\simeq\operatorname{Ber}(\pi_{*}(L|_{D}))/\operatorname{Ber}(\pi_{*}{\mathcal{O}}_{D}).$

If ${\mathcal{B}}(L)$ denotes the Berezinian of the derived push-forward of
$L$ then one has, as in the classical case,

${\mathcal{B}}(L_{1}\otimes
L_{2})\otimes{\mathcal{B}}({\mathcal{O}})\simeq{\mathcal{B}}(L_{1})\otimes{\mathcal{B}}(L_{2})\otimes\langle
L_{1},L_{2}\rangle.$

However, in the supercase, for any line bundles $L$ and $L^{\prime}$ of rank
$1|0$, there is a canonical isomorphism

$\alpha_{D}:\operatorname{Ber}(\pi_{*}L|_{D})\rTo{\sim}\operatorname{Ber}(\pi_{*}(L^{\prime}|_{D})),$
(9.2)

induced by any local isomorphism $L\to L^{\prime}$ (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

${\mathcal{B}}(L_{1}\otimes
L_{2})\otimes{\mathcal{B}}({\mathcal{O}})\simeq{\mathcal{B}}(L_{1})\otimes{\mathcal{B}}(L_{2})$
(9.3)

(see [27]).

More explicitly, locally over $S$ we can pick a relative positive divisor $D$
such that $L_{1}L_{2}(D)$, $L_{2}(D)$ and ${\mathcal{O}}_{X}(D)$ 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

$\displaystyle R\pi_{*}({\mathcal{O}}_{X}):$
$\displaystyle[\pi_{*}L_{2}(D)\to\pi_{*}L_{2}(D)|_{E}],$ $\displaystyle
R\pi_{*}(L_{1}):$
$\displaystyle[\pi_{*}L_{1}L_{2}(D)\to\pi_{*}L_{1}L_{2}(D)|_{E}],$
$\displaystyle R\pi_{*}(L_{2}):$
$\displaystyle[\pi_{*}L_{2}(D)\to\pi_{*}L_{2}(D)|_{D}],$ $\displaystyle
R\pi_{*}(L_{1}L_{2}):$
$\displaystyle[\pi_{*}L_{1}L_{2}(D)\to\pi_{*}L_{1}L_{2}(D)|_{D}].$

Using these resolutions we get an isomorphism

$\displaystyle{\mathcal{B}}(L_{1}L_{2})\otimes{\mathcal{B}}({\mathcal{O}}_{X})={\mathcal{B}}(L_{1}L_{2}(D))\otimes{\mathcal{B}}(L_{2}(D))\otimes\operatorname{Ber}^{-1}(\pi_{*}L_{1}L_{2}(D)|_{D})\otimes\operatorname{Ber}^{-1}(\pi_{*}L_{2}(D)|_{E})$
$\displaystyle\rTo{\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{-1}_{D}\otimes\alpha^{-1}_{E}}{\mathcal{B}}(L_{1}L_{2}(D))\otimes{\mathcal{B}}(L_{2}(D))\otimes\operatorname{Ber}^{-1}(\pi_{*}L_{2}(D)|_{D})\otimes\operatorname{Ber}^{-1}(\pi_{*}L_{1}L_{2}(D)|_{E})={\mathcal{B}}(L_{1})\otimes{\mathcal{B}}(L_{2}).$

Note that isomorphism (9.3) can be used to calculate
${\mathcal{B}}(L_{1}L_{2})$ 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

${\mathcal{B}}(\Pi L)\simeq{\mathcal{B}}(L)^{-1}.$

For example, for a line bundle $L$ of rank $1|0$ one has

${\mathcal{B}}(L^{3})\simeq{\mathcal{B}}(L)^{3}/{\mathcal{B}}({\mathcal{O}})^{2}.$

Applying this to $L=\Pi\omega_{X}$, where $X/S$ is a smooth supercurve, one
gets the analog of Mumford isomorphism in the supercase (see [27]),

${\mathcal{B}}(\omega_{X/S}^{3})\simeq{\mathcal{B}}(\omega_{X/S})^{3}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{2}\simeq{\mathcal{B}}(\omega_{X/S})^{5},$
(9.4)

since ${\mathcal{B}}(L)\simeq{\mathcal{B}}(\omega_{X/S}\otimes L^{-1})$ by
Grothendieck-Serre duality. Thus, denoting
$\operatorname{Ber}_{i}:={\mathcal{B}}(\omega_{X/S}^{i})$, we can write this
isomorphism as

$\operatorname{Ber}_{3}\simeq\operatorname{Ber}_{1}^{5}.$

We will need an extension of isomorphism (9.3) 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 ${\mathcal{F}}$ over $X$,
flat over $S$, we denote by ${\mathcal{B}}({\mathcal{F}})$ the Berezinian of
the perfect complex $R\pi_{*}({\mathcal{F}})$.

######  Lemma 9.2.

Let $\pi:X\to S$ be a stable supercurve, ${\mathcal{F}}$ a coherent sheaf on
$X$ flat over $S$. Assume that ${\mathcal{F}}$ 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

${\mathcal{B}}({\mathcal{F}}\otimes
L)\simeq{\mathcal{B}}({\mathcal{F}})\otimes{\mathcal{B}}(L)\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-1}.$
(9.5)

###### Proof.

We can use the similar recipe as above to construct this isomorphism locally
over $S$: we simply replace $L_{1}$ by ${\mathcal{F}}$, $L_{2}$ 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 $\pi$. ∎

###  9.3. Behavior of the super Mumford isomorphism near the NS boundary
divisor

Now we want to look at the behavior of the super Mumford isomorphism (9.4)
near the generic point of boundary divisors. In the case of a Ramond node it
extends to an isomorphism, since $\omega_{X/S}$ 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. 7.2, so $S$ is a
formal super-polydisk with a special coordinate $t$, $S_{0}\subset S$ is the
divisor $t=0$, $q:S_{0}\to X_{0}\subset X$ is the relative node over $S_{0}$.

Let $S^{\prime}=S\setminus S_{0}$, and let $X^{\prime}=\pi^{-1}(S^{\prime})$
be the corresponding family of smooth supercurves. First, we need to explain
how to extend the line bundles
${\mathcal{B}}(\omega_{X^{\prime}/S^{\prime}}^{3})$ and
${\mathcal{B}}({\mathcal{O}}_{X^{\prime}})$ to $S$. For the second one this is
straightforward: the extension is given by ${\mathcal{B}}({\mathcal{O}}_{X})$.

For any integer $n$ let us set

$\omega_{X/S}^{n}:=j_{*}\omega_{U/S}^{n},$

where $U\subset X$ is the smooth locus of $\pi$. Recall that
$\omega_{X/S}^{2}$ is a line bundle on $X$ (see Theorem 5.1). Thus, we have

$\omega_{X/S}^{3}=j_{*}\omega_{U/S}^{3}\simeq\omega_{X/S}^{2}\otimes
j_{*}\omega_{U/S}\simeq\omega_{X/S}^{2}\otimes\omega_{X/S}.$

This is a coherent sheaf, flat over $S$, so we can take
${\mathcal{B}}(\omega_{X/S}^{3})$ as the desired extension of
${\mathcal{B}}(\omega_{X^{\prime}/S^{\prime}}^{3})$.

Note that by Grothendieck-Serre duality, we have an isomorphism of line bundes
on $S$,

${\mathcal{B}}(\omega_{X/S})\simeq{\mathcal{B}}({\mathcal{O}}_{X}).$

Since $\omega_{X/S}$ is a coherent sheaf flat over $S$, we deduce from Lemma
9.2 an isomorphism

${\mathcal{B}}(\Pi\omega_{X/S}^{3})\simeq{\mathcal{B}}(\Pi\omega_{X/S})\otimes{\mathcal{B}}(\omega_{X/S}^{2})\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-1}\simeq{\mathcal{B}}(\omega_{X/S}^{2})\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-2},$

or equivalently,

${\mathcal{B}}(\omega_{X/S}^{3})\simeq{\mathcal{B}}(\omega_{X/S}^{2})^{-1}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{2}.$
(9.6)

Thus, our problem reduces to studying the behavior of the isomorphism on
$X^{\prime}$,

${\mathcal{B}}(\omega_{X^{\prime}/S^{\prime}}^{2})\simeq{\mathcal{B}}(\Pi\omega_{X^{\prime}/S^{\prime}})^{2}\otimes{\mathcal{B}}({\mathcal{O}}_{X^{\prime}})^{-1}\simeq{\mathcal{B}}({\mathcal{O}}_{X^{\prime}})^{-3}$

near the divisor $S_{0}\subset S$. To this end we look more carefully at the
recipe for this isomorphism outlined in Sec. 9.2.

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

######  Lemma 9.3.

There exists an exact sequence of ${\mathcal{O}}_{X}$-modules,

$0\to\omega^{\operatorname{reg}}_{X/S}\to\omega_{X/S}\to
q_{*}{\mathcal{O}}_{S_{0}}\to 0$

such that in the formal neighborhood of $q(S_{0})$,
$\omega^{\operatorname{reg}}_{X/S}$ is the ${\mathcal{O}}_{X}$-submodule
generated by the sections $s_{1}$ and $s_{2}$ given by (7.11). Furthermore, we
can choose an even section $s$ of $\Pi\omega^{\operatorname{reg}}_{X/S}(D)$ 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

$z_{1}\mapsto u^{-1}z_{1},\ \ z_{2}\mapsto uz_{2},\ \
\theta_{1}\mapsto\theta_{1},\ \ \theta_{2}\mapsto u\theta_{2}$

we can achieve that $u=1$.

###### Proof.

In the formal neighborhood of $q$, the quotient of
$\omega_{X/S}/({\mathcal{O}}_{X}s_{1}+{\mathcal{O}}_{X}s_{2})$ is generated by
$s_{0}$ and is isomorphic to $q_{*}{\mathcal{O}}_{S_{0}}$. This implies the
first assertion. Furthermore, using Lemma 7.4, we see that
$\omega^{\operatorname{reg}}_{X/S}\otimes q_{*}{\mathcal{O}}_{S_{0}}$ is a
free $q_{*}{\mathcal{O}}_{S_{0}}$-module generated by the images of $s_{1}$
and $s_{2}$. For sufficiently positive $D$, the map

$H^{0}(X,\omega^{\operatorname{reg}}_{X/S}(D))\to
H^{0}(X,\omega^{\operatorname{reg}}_{X/S}\otimes q_{*}{\mathcal{O}}_{S_{0}})$

is surjective, so we can choose an odd section $s\in
H^{0}(X,\omega^{\operatorname{reg}}_{X/S}(D))$ such that

$s\equiv s_{1}+s_{2}\mod J_{q(S_{0})}.$

Thus, in the formal neighborhood of $q$, we can write
$s=u_{1}s_{1}+u_{2}s_{2}$ 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. ∎

Now we define coherent sheaves ${\mathcal{F}}$ and ${\mathcal{G}}$ on $X$ by
the exact sequences

$0\to{\mathcal{O}}_{X}\rTo{s}\Pi\omega_{X/S}(D)\to{\mathcal{F}}\to 0,$
$0\to\Pi\omega_{X/S}\rTo{\cdot s}\omega_{X/S}^{2}(D)\to{\mathcal{G}}\to 0.$

Let $\hat{{\mathcal{F}}}$ and $\hat{{\mathcal{G}}}$ denote the completions of
${\mathcal{F}}$ and ${\mathcal{G}}$ at $q$

######  Lemma 9.4.

(a) The space $\hat{{\mathcal{F}}}$ is a free ${\mathcal{O}}_{S}$-module with
the basis $s_{1},s_{0}$.

(b) We have an isomorphism $\hat{{\mathcal{G}}}\simeq{\mathcal{O}}_{Z}$, where
$Z\subset X$ is a sub-superscheme given by

$z_{1}=-t,\ \ z_{2}=t,\ \ \theta_{1}+\theta_{2}=0.$

In particular, $\hat{{\mathcal{G}}}$ is a free ${\mathcal{O}}_{S}$-module with
the basis $e,\theta_{1}e$, where $e$ is the local generator of
$\omega_{X/S}^{2}$ given by (7.12). The sheaf $\hat{{\mathcal{F}}}$ is also
scheme-theoretically supported on $Z$.

(c) The Berezinian of the map of ${\mathcal{O}}_{S}$-modules

$s_{1}\cdot?:\hat{{\mathcal{F}}}\to\hat{{\mathcal{G}}}$

is equal to $ft$, where $f$ is an invertible function.

###### Proof.

(a) Lemma 7.4 immediately shows that the map

${\mathcal{O}}_{S}\to\Pi\omega_{X/S}:f\mapsto f\cdot(s_{1}+s_{2})$

is injective. Furthermore, the submodule ${\mathcal{O}}_{X}(s_{1}+s_{2})$ has
the following topological basis over ${\mathcal{O}}_{S}$:

$\displaystyle s_{1}+s_{2},\ \
z_{1}^{n+1}(s_{1}+s_{2})=z_{1}^{n}(z_{1}+t)s_{1},\ \ z_{2}^{n}(z_{2}-t)s_{2},\
\ \theta_{1}(s_{1}+s_{2})=\theta_{1}s_{1}+ts_{0},$ $\displaystyle
z_{1}^{n+1}\theta_{1}(s_{1}+s_{2})=z_{1}^{n}(z_{1}+t)\theta_{1}s_{1},\ \
\theta_{2}(s_{1}+s_{2})=\theta_{2}s_{2}+ts_{0},\ \
z_{2}^{n}(z_{2}-t)\theta_{2}s_{2},$

where $n\geq 0$. Hence, the quotient by this submodule has the images of
$s_{0},s_{1}$ as a basis over ${\mathcal{O}}_{S}$.

(b) Near the node, $\omega_{X/S}^{2}$ is a free ${\mathcal{O}}_{X}$-module
with one even generator $e$. Furthermore, it is easy to check that with
respect to the map $\omega_{X/S}^{\otimes 2}\to\omega_{X/S}^{2}$ one has

$s_{1}^{2}=z_{1}e,\ \ s_{2}^{2}=-z_{2}e,\ \ s_{1}s_{2}=te,\ \
s_{0}s_{i}=\theta_{i}e,$

for $i=1,2$. Thus, the image of the multiplication by $s_{1}+s_{2}$ is the
${\mathcal{O}}_{X}$-submodule generated by

$s_{1}(s_{1}+s_{2})=(z_{1}+t)e,\ \ s_{2}(s_{1}+s_{2})=(t-z_{2})e,\ \
s_{0}(s_{1}+s_{2})=(\theta_{1}+\theta_{2})e.$

In other words, we have an identification

$\hat{{\mathcal{G}}}\simeq{\mathcal{O}}_{X}/(z_{1}+t,z_{2}-t,\theta_{1}+\theta_{2})={\mathcal{O}}_{Z}\simeq{\mathcal{O}}_{S}[\theta_{1}],$

as claimed.

(c) The operator of multiplication by $s_{1}$ acts on the bases of
$\hat{{\mathcal{F}}}$ and $\hat{{\mathcal{G}}}$ as

$s_{1}\mapsto s_{1}^{2}=-te,\ \ s_{0}\mapsto s_{1}s_{0}=\theta_{1}e,$

Hence, the Berezinian is equal to $-t$ (recall that $s_{1}$ is an even
generator of $\hat{{\mathcal{F}}}$, while $s_{0}$ is an odd generator). ∎

Let us consider the following resolutions for the derived push-forwards under
$\pi$:

$\displaystyle R\pi_{*}({\mathcal{O}}_{X}):$
$\displaystyle[\pi_{*}(\Pi\omega_{X/S}(D)\to\pi_{*}{\mathcal{F}}],$
$\displaystyle R\pi_{*}(\Pi\omega_{X/S}):$
$\displaystyle[\pi_{*}(\omega_{X/S}^{2}(D))\to\pi_{*}{\mathcal{G}}],$
$\displaystyle R\pi_{*}(\Pi\omega_{X/S}):$
$\displaystyle[\pi_{*}(\Pi\omega_{X/S}(D))\to\pi_{*}(\Pi\omega_{X/S}(D)|_{D})],$
$\displaystyle R\pi_{*}(\omega_{X/S}^{2}):$
$\displaystyle[\pi_{*}(\omega_{X/S}^{2}(D))\to\pi_{*}(\omega_{X/S}^{2}(D)|_{D})].$

These resolutions give a canonical isomorphism over $S$,

${\mathcal{B}}(\omega_{X/S}^{2})\otimes{\mathcal{B}}({\mathcal{O}}_{X})\simeq\operatorname{Ber}(R\pi_{*}(\Pi\omega_{X/S}))^{\otimes
2}\otimes\operatorname{Ber}(\pi_{*}{\mathcal{G}})\otimes\operatorname{Ber}(\pi_{*}{\mathcal{F}})^{-1}.$
(9.7)

Now the super Mumford isomorphism for the induced smooth family
$X_{S^{\prime}}\to S^{\prime}$ is obtained by choosing an isomorphism of

${\mathcal{F}}|_{X_{S^{\prime}}}\simeq{\mathcal{G}}|_{X_{S^{\prime}}}$

(where both sheaves are supported on the zero divisor of $s$ in
$X_{S^{\prime}}$) and using the Berezinian of the induced map on $R\pi_{*}$ to
get an isomorphism of $\operatorname{Ber}(\pi_{*}{\mathcal{F}})|_{S^{\prime}}$
with $\operatorname{Ber}(\pi_{*}{\mathcal{G}})|_{S^{\prime}}$ (see (9.2)). For
this we are going to construct a morphism

$\alpha:{\mathcal{F}}\to{\mathcal{G}},$

which is an isomorphism away from the node, and use its restriction to
$X_{S^{\prime}}$.

Note that both ${\mathcal{F}}$ and ${\mathcal{G}}$ 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 $E$,

${\mathcal{F}}\simeq{\mathcal{F}}_{Z}\oplus{\mathcal{F}}_{E},\ \
{\mathcal{G}}\simeq{\mathcal{G}}_{Z}\oplus{\mathcal{G}}_{E}.$

We can choose separately morphisms ${\mathcal{F}}_{E}\to{\mathcal{G}}_{E}$ and
${\mathcal{F}}_{Z}\to{\mathcal{G}}_{Z}$.

We have

${\mathcal{F}}_{E}\simeq\pi_{*}(\Pi\omega_{X/S}(D)|_{E}),\ \
{\mathcal{G}}_{E}\simeq\pi_{*}(\omega_{X/S}^{2}(D)|_{E}),$

so both are isomorphic to ${\mathcal{O}}_{E}$, and we choose any isomorphism
between them as a morphism $\alpha_{E}:{\mathcal{F}}_{E}\to{\mathcal{G}}_{E}$.
On the other hand, we have

${\mathcal{F}}_{Z}\simeq\hat{F},\ \
{\mathcal{G}}_{Z}\simeq\hat{{\mathcal{G}}},$

Hence, we can use the morphism of ${\mathcal{O}}_{X}$-modules,

$\mu_{s_{1}}:\hat{{\mathcal{F}}}\to\hat{{\mathcal{G}}},$

given by the multiplication with $s_{1}$, as our morphism
${\mathcal{F}}_{Z}\to{\mathcal{G}}_{Z}$. By Lemma 9.4, it restricts to an
isomorphism over $t\neq 0$. Thus, we get the desired morphism

$\alpha=(\mu_{s^{\prime}},\alpha_{E}):{\mathcal{F}}\to{\mathcal{G}}.$

By definition, the super Mumford isomorphism for $X_{S^{\prime}}\to
S^{\prime}$ is obtained by restricting the isomorphism of ilne bundles (9.7)
to $S^{\prime}$ and multiplying it with
$\operatorname{ber}R\pi_{*}(\alpha|_{X_{S^{\prime}}})^{-1}\in{\mathcal{B}}({\mathcal{F}})\otimes{\mathcal{B}}({\mathcal{G}})^{-1}$.
Since $\operatorname{ber}R\pi_{*}(\alpha_{E^{\prime}})$ is invertible on $S$,
we see that

$\operatorname{ber}R\pi_{*}(\alpha|_{X_{S^{\prime}}})=u\cdot\operatorname{ber}\pi_{*}(\mu_{s_{1}}),$

where $u$ is an invertible function on $S$. Using Lemma 9.4(iii), we deduce
that

$\operatorname{ber}R\pi_{*}(\alpha|_{X_{S^{\prime}}})^{-1}=u^{\prime}\cdot
t^{-1},$

for some invertible function $u^{\prime}$ on $S$, with respect to some bases
of ${\mathcal{B}}({\mathcal{F}})$ and ${\mathcal{B}}({\mathcal{G}})$ on $S$.

Thus, we obtained an isomorphism

${\mathcal{B}}(\omega_{X/S}^{2})\rTo{\sim}{\mathcal{B}}({\mathcal{O}}_{X})^{-1}\otimes\operatorname{Ber}(R\pi_{*}(\Pi\omega_{X/S}))^{\otimes
2}(\Delta),$ (9.8)

where $\Delta$ is the divisor given by $t=0$. Combining this with isomorphism
(9.6), we get the following local result.

######  Proposition 9.5.

In the above situation the super Mumford isomorphism

${\mathcal{B}}(\omega_{X^{\prime}/S^{\prime}}^{-2})\simeq{\mathcal{B}}(\omega_{X^{\prime}/S^{\prime}}^{3})\simeq{\mathcal{B}}({\mathcal{O}}_{X^{\prime}})^{5}$

extends to an isomorphism

${\mathcal{B}}(\omega_{X/S}^{-2})\simeq{\mathcal{B}}(\omega_{X/S}^{3})\simeq{\mathcal{B}}({\mathcal{O}}_{X})^{5}(-\Delta).$

Combining this with Corollary 7.7 and with Proposition 7.9 we obtain the
following result.

######  Theorem 9.6.

Let $\overline{\SS}_{g}$ denote the moduli stack of stable supercurves of
genus $g$, and let $\Delta_{NS}\subset\overline{\SS}_{g}$ (resp.,
$\Delta_{R}\subset\overline{\SS}_{g}$) denote the boundary divisor supported
on the locus where a supercurve acquires an NS node (resp., a Ramond node),
defined in Sec. 8.2. Then one has an isomorphism of line bundles

$K_{\overline{\SS}_{g}}\simeq\operatorname{Ber}_{1}^{5}(-2\Delta_{NS}-\Delta_{R}).$
(9.9)

###### Proof.

Let $\pi:X\to\overline{\SS}_{g}$ denote the universal supercurve. First, we
have an isomorphism

$K_{\overline{\SS}_{g}}\simeq\operatorname{Ber}^{-1}R^{1}\pi_{*}\omega_{X/\overline{\SS}_{g}}^{-2}(-\Delta_{NS}-\Delta_{R})\simeq{\mathcal{B}}(\omega_{X/\overline{\SS}_{g}}^{-2})(-\Delta_{NS}-\Delta_{R})$

(see Corollary 7.7 and Proposition 7.9). Next, by Proposition 9.5, we have

${\mathcal{B}}(\omega_{X/\overline{\SS}_{g}}^{-2})\simeq{\mathcal{B}}(\omega_{X/\overline{\SS}_{g}}^{3})\simeq\operatorname{Ber}_{1}^{5}(-\Delta_{NS})$

(recall that the super-Mumford isomorphism (9.4) extends over the Ramond type
boundary divisor $\Delta_{R}$). Combining this with the previous isomorphism
we get the result. ∎

###  9.4. Canonical line bundle on the moduli space of stable supercurves with
punctures

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 $D_{i}\subset X$ the
divisors associated with $P_{i}$ as in Sec. 2.4.

For every $i=1,\ldots,m$, let us define a line bundle of rank $0|1$ on $S$,

$\Psi_{i}:=P_{i}^{*}\omega_{X/S}.$

######  Lemma 9.7.

One has natural isomorphisms of line bundles of rank $0|1$ on $S$,

$\Psi_{i}\simeq\pi_{*}{\mathcal{O}}_{D_{i}}/{\mathcal{O}}_{S},$
$\Psi_{i}^{-1}\simeq P_{i}^{*}\omega_{X/S}(D_{i})\simeq
P_{i}^{*}\omega_{D_{i}/S}\simeq\operatorname{Ber}(\pi_{*}{\mathcal{O}}_{D_{i}}).$

Hence, for any line bundle $L$ of rank $1|0$ on $X$, one has

$\operatorname{Ber}(\pi_{*}L|_{D_{i}})\simeq\Psi_{i}^{-1},$

while for a line bundle $M$ of rank $0|1$, one has

$\operatorname{Ber}(\pi_{*}M|_{D_{i}})\simeq\Psi_{i}.$

###### Proof.

Let us set $\Psi=\Psi_{i}$ and

$\widetilde{\Psi}:=\pi_{*}{\mathcal{O}}_{D}/{\mathcal{O}}_{S},$

where $D=D_{i}$ and $P=P_{i}$. Then $\widetilde{\Psi}$ is a line bundle of
rank $0|1$, and we have

$\operatorname{Ber}(\pi_{*}{\mathcal{O}}_{D})\simeq\widetilde{\Psi}^{-1}.$

Hence, from (9.2) we get the last two isomorphisms with $\Psi_{i}$ replaced by
$\widetilde{\Psi}$.

Applying this to the line bundle $\omega_{X/S}^{-2}$ of rank $1|0$, we get an
isomorphism

$\operatorname{Ber}(\pi_{*}\omega_{X/S}^{-2}|_{D})\simeq\widetilde{\Psi}^{-1}.$

But from Lemma 2.9, we have an isomorphism over $S$,

$\pi_{*}(\omega_{X/S}^{-2}|_{D})\simeq{\mathcal{T}}_{X/S}|_{P}.$

Hence, passing to the Berezinians we get

$\widetilde{\Psi}^{-1}\simeq\omega_{X/S}^{-1}|_{P},$

i.e., $\widetilde{\Psi}\simeq\Psi$.

Next, let $I_{D}\subset I_{P}\subset{\mathcal{O}}_{X}$ 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

$\pi_{*}{\mathcal{O}}_{D}={\mathcal{O}}_{S}\oplus\pi_{*}(I_{P}/I_{D}),$

so

$\Psi\simeq\widetilde{\Psi}\simeq\pi_{*}(I_{P}/I_{D}).$

Note also that $I_{P}^{2}\subset I_{D}$, so $I_{P}/I_{D}$ can be identified
with the conormal sheaf to $P$ in $D$. Now, since
$\omega_{P/S}={\mathcal{O}}_{S}$, the exact sequence

$0\to\Psi\to\Omega_{D/S}|_{P}\to{\mathcal{O}}_{P}\to 0$

gives an isomorphism

$P^{*}\omega_{D/S}\simeq\operatorname{Ber}(\Psi)=\Psi^{-1}.$

∎

######  Corollary 9.8.

The line bundle $P_{i}^{*}\omega_{X/S}^{2}(D_{i})$ on $S$ is canonically
trivialized.

######  Example 9.9.

Suppose $X\to S$, $(P_{i})$, is a family of supercurves with NS-punctures over
an even base $S$. Then ${\mathcal{O}}_{X}={\mathcal{O}}_{C}\oplus L$, where
$(C,L)$ is the underlying family of curves with spin-structures, and
${\mathcal{I}}_{P_{i}}={\mathcal{I}}_{p_{i}}\oplus L$, where $p_{i}\subset C$
are marked points on $C$. We also have
${\mathcal{I}}_{D}={\mathcal{I}}_{p_{i}}\otimes_{{\mathcal{O}}_{C}}{\mathcal{O}}_{X}$.
Thus,

$\Psi_{i}^{-1}=P_{i}^{*}(\omega_{X/S}(D_{i}))=P_{i}^{*}(L(p_{i})\oplus\omega_{C/S}(p_{i}))\simeq
p_{i}^{*}L(p_{i})\simeq p_{i}^{*}L^{-1},$

where the last isomorphisms is induced by the trivialization of
$L^{2}(p_{i})|_{p_{i}}\simeq\omega_{C/S}(p_{i})|_{p_{i}}$.

######  Remark 9.10.

Note that for an NS-puncture $P_{i}$ and the corresponding divisor $D_{i}$,
the decomposition (2.5) can be rewritten as

$\pi_{*}{\mathcal{O}}_{D_{i}}={\mathcal{O}}_{S}\oplus\Psi_{i},$

where $\Psi_{i}$ is a square-zero ideal. For any line bundle $L$ over $D_{i}$
we have an exact sequence

$0\to P_{i_{*}}(\Psi_{i}\otimes P_{i}^{*}L)\to L\to P_{i*}P_{i}^{*}L\to 0$

In the case $L=\omega_{X/S}(D_{i})|_{D_{i}}=\omega_{D_{i}/S}$, the induced
exact sequence of push-forwards to $S$ has a splitting

$\pi_{*}(\omega_{X/S}(D_{i})/\omega_{X/S})\to{\mathcal{O}}_{S}$

given by the residue map for the $0|1$-dimensional superscheme $D_{i}/S$, so
we have a decomposition

$\pi_{*}(\omega_{D_{i}/S})=\Psi_{i}^{-1}\oplus{\mathcal{O}}_{S}.$

For every Ramond divisor $R_{j}\subset X$ let us consider the line bundle on
the base,

$\Phi_{j}:=\operatorname{Ber}(\pi_{*}{\mathcal{O}}_{R_{j}})^{-1}\simeq\operatorname{Ber}(\pi_{*}\omega_{R_{j}}).$

Note that $\Phi_{j}=\Phi_{R_{j}}$ (see (8.8)), so the line bundles
$\Phi_{j}^{2}$ are canonically trivialized.

Let us consider the line bundle of rank $1|0$ over $X$,

${\mathcal{L}}(X,P_{\bullet},R_{\bullet}):=\omega_{X/S}^{-2}(-\sum_{i}D_{i}-2\sum_{j}R_{j}).$

######  Lemma 9.11.

One has a natural isomorphism

${\mathcal{B}}({\mathcal{L}}(X,P_{\bullet},R_{\bullet}))\simeq{\mathcal{B}}(\omega_{X/S}^{3})\otimes\bigotimes_{i}\Psi_{i}$

###### Proof.

Let us set $D=\sum_{i}D_{i}$, $R=\sum_{j}R_{j}$. First, the Grothendieck
duality gives an isomorphism

${\mathcal{B}}({\mathcal{L}}(X,P_{\bullet},R_{\bullet}))\simeq{\mathcal{B}}(\omega_{X/S}^{3}(D+2R)).$

Now using the exact sequence

$0\to\omega_{X/S}^{3}(2R)\to\omega_{X/S}^{3}(2R)(D)\to\omega_{X/S}^{3}(2R)(D)|_{D}\to
0,$ (9.10)

we get using (9.2) and Lemma 9.7,

${\mathcal{B}}(\omega_{X/S}^{3}(D+2R))\simeq{\mathcal{B}}(\omega_{X/S}^{3}(2R))\otimes\operatorname{Ber}(\pi_{*}\omega_{X/{\mathcal{M}}}^{3}(D)|_{D})\simeq{\mathcal{B}}(\omega_{X/M}^{3}(2R))\otimes\bigotimes_{i}\Psi_{i}.$

Similarly, using the fitration of $\omega_{X/S}^{3}(2R)/\omega_{X/S}^{3}$ with
subquotients $\omega_{X/S}^{3}(2R)|_{R}$ and $\omega_{X/S}^{3}(R)|_{R}$, we
get an isomorphism

${\mathcal{B}}(\omega_{X/S}^{3}(2R))\simeq{\mathcal{B}}(\omega_{X/S}^{3})\otimes\bigotimes_{j}\Phi_{j}^{2}.$

It remains to we use the trivializations of $\Phi_{j}^{2}$. ∎

Now the same argument as in Theorem 9.6 gives the formula for the canonical
line bundle on the moduli stack of stable supercurves with punctures.

###### Proof of Theorem B.

Let $\overline{\SS}=\overline{\SS}_{g,n_{NS},n_{R}}$. We combine the
isomorphism

$K_{\overline{\SS}}\simeq\operatorname{Ber}^{-1}R^{1}\pi_{*}{\mathcal{L}}(X,P_{\bullet},R_{\bullet})(-\Delta_{NS}-\Delta_{R})\simeq{\mathcal{B}}({\mathcal{L}}(X,P_{\bullet},R_{\bullet}))(-\Delta_{NS}-\Delta_{R})$

(see Sec. 7.4) with Lemma 9.11 and the isomophism

${\mathcal{B}}(\omega_{X/\overline{\SS}}^{3})\simeq\operatorname{Ber}_{1}^{5}(-\Delta_{NS}).$

This gives the required isomorphism

$K_{\overline{\SS}}\simeq\operatorname{Ber}_{1}^{5}\otimes\bigotimes_{i=1}^{m}\Psi_{i}(-2\Delta_{NS}-\Delta_{R}),$
(9.11)

∎

##  10\. Splitting at the boundary divisor

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. 8.2) 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.

###  10.1. NS boundary components

Let $\iota:B\to\overline{\SS}$ be the standard gluing map covering an NS type
boundary component, i.e., one of the maps (8.6) or (8.7), 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 $\widetilde{X}\to B$ into a node $Q\subset X_{B}$. Let
$D_{1},D_{2}\subset\widetilde{X}$ be the corresponding divisors. Note that we
have a finite morphism $\rho:\widetilde{X}\to X_{B}$, and an exact sequence on
$X_{B}$,

$0\to{\mathcal{O}}_{X_{B}}\to\rho_{*}{\mathcal{O}}_{\widetilde{X}}\to{\mathcal{O}}_{Q}\to
0.$

In particular, the Berezinian line bundle
${\mathcal{B}}({\mathcal{O}}_{X_{B}})$ for the family $X_{B}\to B$ is
naturally identified with ${\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})$
defined for the family $\widetilde{X}\to B$.

Note that in the case of a separating node, where $B=\SS_{1}\times\SS_{2}$,
the line bundle ${\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})$ is the exterior
product of two similar line bundles on the factors $\SS_{1}$ and $\SS_{2}$.

We can rewrite (9.11) near $B$ as an isomorphism

$K_{\overline{\SS}}(2\Delta_{NS})\simeq{\mathcal{B}}({\mathcal{O}}_{X})^{5}.$

Thus, pulling it back to $B$ leads to an isomorphism

$K_{B}\otimes N_{B}\simeq{\mathcal{B}}({\mathcal{O}}_{X_{B}})^{5},$

where $N_{B}$ is the normal bundle defined as the pull-back of
${\mathcal{O}}(\Delta_{NS})$ to $B$. Note that the universal supercurve over
$B$ is equipped with two NS punctures $P_{1},P_{2}$ and the isomorphism (9.11)
in smaller genus gives

$K_{B}\simeq{\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})^{5}\otimes\Psi_{1}\otimes\Psi_{2}.$

Comparing with the previous isomorphism we get an isomorphism

$N_{B}\simeq\Psi_{1}^{-1}\otimes\Psi_{2}^{-1}.$

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 $\omega_{X/S}^{2}$ defined on the
universal curve of $\overline{\SS}$ by extending from the smooth locus (see
Theorem 5.1).

######  Lemma 10.1.

(i) For any integer $m$ let us set

$\omega_{X_{B}/B}^{2m}:=\omega_{X/\overline{\SS}}^{2m}|_{B}.$

Then one has an exact sequence on $X_{B}$,

$0\to\omega_{X_{B}/B}^{2m}\to\rho_{*}\omega_{\widetilde{X}/B}^{2m}(mD_{1}+mD_{2})\to{\mathcal{O}}_{Q}\to
0$ (10.1)

where $Q\subset X_{B}$ is the relative node obtained by gluing $P_{1}\simeq
B\simeq P_{2}$. Here we use the canonical trivializations of the restrictions
of $\omega_{\widetilde{X}/B}^{2}(D_{1}+D_{2})$ to $P_{1}$ and $P_{2}$ (see
Corollary 9.8). Hence, we have a natural isomorphism

${\mathcal{B}}(\omega_{X/\overline{\SS}}^{2m})|_{B}\simeq{\mathcal{B}}(\omega_{\widetilde{X}/B}^{2m}(mD_{1}+mD_{2})).$
(10.2)

(ii) Let us set
$\omega_{X_{B}/B}^{2m+1}:=\omega_{X_{B}/B}^{2m}\otimes\omega_{X_{B}/B}$. Then
one has an exact sequence on $X_{B}$,

$0\to\rho_{*}\omega^{2m+1}_{\widetilde{X}/B}(mD_{1}+mD_{2})\to\omega_{X_{B}/B}^{2m+1}\to
Q_{*}{\mathcal{O}}_{S}\to 0.$ (10.3)

Hence, we have a natural isomorphism

${\mathcal{B}}(\omega_{X/\overline{\SS}}^{2m+1})|_{B}\simeq{\mathcal{B}}(\omega_{\widetilde{X}/B}^{2m+1}(mD_{1}+mD_{2})).$
(10.4)

In addition, we have a natural exact sequence

$0\to\omega_{X_{B}/B}^{2m+1}\to\rho_{*}\omega_{\widetilde{X}/B}^{2m+1}((m+1)D_{1}+(m+1)D_{2})\to{\mathcal{C}}\to
0,$

where ${\mathcal{C}}$ is a sheaf supported on the node fitting into an exact
sequence

$0\to Q_{*}{\mathcal{O}}_{S}\to{\mathcal{C}}\to
Q_{*}(\Psi_{1}^{-1}\oplus\Psi_{2}^{-1})\to 0.$ (10.5)

(iii) One has an exact sequence on $X_{B}$,

$0\to{\mathcal{K}}\to\Omega_{X_{B}/B}\to\rho_{*}\Omega_{\widetilde{X}/B}\to
0,$

where the sheaf ${\mathcal{K}}$ 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, $\operatorname{Ber}\pi_{*}{\mathcal{K}}$ is canonically
trivial, so

$\operatorname{Ber}R\pi_{*}(\Omega_{X_{B}/B})\simeq\operatorname{Ber}R\pi_{*}\Omega_{\widetilde{X}/B}.$

(iv) One has an exact sequence over the smooth locus of $B$,

$0\to
j_{*}\Omega_{U/\overline{\SS}}|_{X_{B}}\to\Omega_{\widetilde{X}/B}(D_{1}+D_{2})\to\widetilde{{\mathcal{C}}}\to
0,$

where $U\subset X$ is the smooth locus of $X\to\overline{\SS}$, and the sheaf
$\widetilde{{\mathcal{C}}}$ is supported on the node and has a filtration with
the subfactors

${\mathcal{O}}_{Q},\ {\mathcal{O}}_{Q},\
Q_{*}(\Psi_{1}^{-1}\oplus\Psi_{2}^{-1}).$

###### Proof.

(i) We just have to identify the pull-back of the line bundle
$\omega_{X_{B}/B}^{2}$ to $\widetilde{X}$, $\rho^{*}\omega_{X_{B}/B}^{2}$ with
$\omega_{\widetilde{X}/B}^{2}(D_{1}+D_{2})$. Note that we have a natural
identification of these line bundles over the smooth locus of $\widetilde{X}$.
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. 7.2 and use a
generator $e$ of $\omega_{X/S}^{2}$ (see (7.12)). Since $\rho^{*}e$ is a
generator of $\omega_{\widetilde{X}/B}^{2}(D_{1}+D_{2})$, the assertion
follows.

(ii) First, let us consider the case $m=0$. Using Lemma 7.4 we easily see that
there is an injective map
$\omega_{X_{B}/B}\to\rho_{*}\omega_{\widetilde{X}/B}(D_{1}+D_{2})$, and that
$\omega_{X_{B}/B}$ contains $\rho_{*}\omega_{\widetilde{X}/B}$. Thus, we have

${\mathcal{C}}:=\operatorname{coker}(\omega_{X_{B}/B}\to\rho_{*}\omega_{\widetilde{X}/B}(D_{1}+D_{2}))=\operatorname{coker}(\omega_{X_{B}/B}/\omega_{\widetilde{X}/B}\to\rho_{*}(\omega_{\widetilde{X}/B}(D_{1}+D_{2})/\omega_{\widetilde{X}/B})).$

Note that since $\Psi_{i}^{-1}\simeq
P_{i}^{*}\omega_{\widetilde{X}/B}(D_{i})$, by Remark 9.10, we have an exact
sequence

$0\to{\mathcal{O}}_{P_{1}}\oplus{\mathcal{O}}_{P_{2}}\to\omega_{\widetilde{X}/B}(D_{1}+D_{2})/\omega_{\widetilde{X}/B}\to
P_{1*}\Psi_{1}^{-1}\oplus P_{2*}\Psi_{2}^{-1}\to 0$

In local coordinates, the projection to $P_{i*}\Psi_{i}^{-1}$ sends
$\frac{1}{z_{i}}[dz_{i}|d\theta_{i}]$ to a generator, and sends
$\frac{\theta_{i}}{z_{i}}[dz_{i}|d\theta_{i}]$ to zero. In particular, this
projection vanishes on the image of
$\omega_{X_{B}/B}/\omega_{\widetilde{X}/B}$. Thus, we have an embedding

$\omega_{X_{B}/B}/\omega_{\widetilde{X}/B}\hookrightarrow{\mathcal{O}}_{Q}\oplus{\mathcal{O}}_{Q}.$

We claim that $\omega_{X_{B}/B}/\omega_{\widetilde{X}/B}$ coincides with the
kernel of the addition map
${\mathcal{O}}_{Q}\oplus{\mathcal{O}}_{Q}\to{\mathcal{O}}_{Q}$. Indeed, this
is a local statement. In local coordinates the generators of two summands
${\mathcal{O}}_{Q}$ are $\frac{\theta_{i}}{z_{i}}[dz_{i}|d\theta_{i}]$, and
the generator $s_{0}$ of $\omega_{X_{B}/B}/\omega_{\widetilde{X}/B}$ is mapped
to their difference.

From this we also see that ${\mathcal{C}}$ fits into the exact sequence
(10.5).

To derive the case of arbitrary $m$ from that of $m=0$, we tensor the sequence
(10.5) with the line bundle $\omega^{2m}_{X_{B}/B}$. By the triviality of
$Q^{*}(\omega_{X_{B}/B}^{2}$, we get the sequence of the required form.

(iii) First, we note that ${\mathcal{K}}$ 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 $\widetilde{X}$ 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

$\rho_{*}(I_{P_{1}}\oplus 0),\rho_{*}(0\oplus
I_{P_{2}})\subset{\mathcal{O}}_{X_{B}}\subset\rho_{*}{\mathcal{O}}_{\widetilde{X}}.$

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

$\kappa:I_{P_{1}}/I_{P_{1}}^{2}\otimes
I_{P_{2}}/I_{P_{2}}^{2}\to{\mathcal{K}}\subset\Omega_{X_{B}/B}:a_{1}\otimes
a_{2})\mapsto(a_{1},0)\cdot d(0,a_{2}).$

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 8.2(ii), one checks that
$\kappa$ is an isomorphism, and the assertion follows.

(iv) This follows from parts (i), (ii) and from the exact sequence (8.1) (and
a similar exact sequence for $\Omega_{\widetilde{X}/B}$ which holds over the
smooth locus). ∎

We can use the definition of the NS boundary line bundle (8.4) to compute the
normal bundle over the smooth part of $B$. From the above Lemma we get

$\operatorname{Ber}R\pi_{*}(j_{*}\Omega_{U/S})|_{B}\simeq\operatorname{Ber}R\pi_{*}(\Omega_{\widetilde{X}/B}(D_{1}+D_{2}))\otimes\Psi_{1}^{-1}\Psi_{2}^{-1},$
$\operatorname{Ber}R\pi_{*}(\Omega_{X/S}|_{B})\simeq\operatorname{Ber}R\pi_{*}\Omega_{\widetilde{X}/B}.$

Finally the exact sequence

$0\to\omega_{\widetilde{X}/B}^{2}\to\Omega_{\widetilde{X}/B}\to\omega_{\widetilde{X}/B}\to
0$

near $D_{1},D_{2}$, together with Lemma 9.7, show that

$\operatorname{Ber}R\pi_{*}\Omega_{\widetilde{X}/B}(D_{1}+D_{2})\simeq\operatorname{Ber}R\pi_{*}\Omega_{\widetilde{X}/B}.$

Hence, we deduce an isomorphism

${\mathcal{O}}(\Delta)|_{B}\simeq\operatorname{Ber}R\pi_{*}(j_{*}\Omega_{U/S})|_{B}\otimes\operatorname{Ber}^{-1}R\pi_{*}(\Omega_{X/S}|_{B})\simeq\Psi_{1}^{-1}\Psi_{2}^{-1}.$
(10.6)

Now we can state a compatibility result between the super Mumford isomorphisms
over $\overline{\SS}$ and over the NS boundary component $B$. Recall that we
have the super Mumford isomorphism

$\mu_{\overline{\SS}}:{\mathcal{B}}(\omega_{X/\overline{\SS}}^{-2})\rTo{\mathcal{B}}({\mathcal{O}}_{X})^{5}(-\Delta)$

near $B$ (see Prop. 9.5). We also have the super Mumford isomorphism for the
family $\widetilde{X}/B$, over the smooth locus:

$\mu_{B}:{\mathcal{B}}(\omega_{\widetilde{X}/B}^{-2}(-D_{1}-D_{2}))\rTo{\sim}{\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})^{5}\otimes\Psi_{1}\otimes\Psi_{2}$

(see Lemma 9.11).

######  Theorem 10.2.

The following diagram of isomorphisms of line bundles on $B$ is commutative up
to a sign

Recall that to get $\mu_{\overline{\SS}}$ we used three isomorphisms: the
Grothendieck-Serre duality isomoprhism

$SD_{X/\overline{\SS}}:{\mathcal{B}}(\omega_{X/\overline{\SS}}^{-2})\rTo{\sim}{\mathcal{B}}(\omega_{X/\overline{\SS}}^{3}),$

the isomorphism

${\mathcal{B}}(\omega_{X/\overline{\SS}}^{3})\simeq{\mathcal{B}}(\omega_{X/\overline{\SS}}^{2})^{-1}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{2},$

and the isomorphism

${\mathcal{B}}(\omega_{X/\overline{\SS}}^{2})\simeq{\mathcal{B}}(\omega_{X/\overline{\SS}})^{-2}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-1}(\Delta)$

(see Sec. 9.3). Similarly, $\mu_{B}$ is a composition of similar three
isomorphisms. So we can reduce the proof of Theorem 10.2 to separate
compatibilities involving each of these three isomorphisms. We deal with this
compatibilities in the next three lemmas.

######  Lemma 10.3.

For any $m\in\operatorname{\mathbb{Z}}$, the following diagram of isomorphisms
between line bundles on $B$ is commutative: where the horizontal arrows are
given by Grothendieck-Serre duality.

###### Proof.

To begin with we can replace the upper horizontal arrow with the one induced
by the Grothendieck-Serre duality for $X_{B}$ over $B$,

${\mathcal{B}}(\omega^{-2m}_{X_{B}/B})\rTo{SD_{X_{B}/B}}{\mathcal{B}}(\omega^{2m+1}_{X_{B}/B}).$

Let us set for brevity

$L:=\omega^{-2m}_{X_{B}/B},\ \
\widetilde{L}:=\omega_{\widetilde{X}/B}^{-2m}(-mD_{1}-mD_{2}).$

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

$SD_{f}:Rf_{*}(F)\rTo{\sim}Rf_{*}(R\underline{\operatorname{Hom}}(F,\omega_{f}[\dim
f]))^{\vee}.$

Then we claim that there is an isomorphism of the exact triangles induced by
the exact sequences (10.1) and (10.3)

$\begin{diagram}$ (10.7)

Clearly this would imply the claimed commutativity.

Now we claim that commutativity of both squares in (10.7) follows from the
general property of Grothendieck-Serre duality for a pair of morphisms
$X\rTo{g}Y\rTo{f}X$, where the lower horizontal arrow is induced by the
Grothendieck-Serre duality isomorphism

$R\underline{\operatorname{Hom}}(Rg_{*}(F),\omega_{f}[\dim f])\simeq
Rg_{*}R\underline{\operatorname{Hom}}(F,\omega_{f\circ g}[\dim(f\circ g)]).$

Indeed, applying this to $f=\pi$, $g=\rho$ and $F=\widetilde{L}$ allows us to
identify the middle vertical arrow in (10.7) with the map

$R\pi_{*}(\rho_{*}\widetilde{L})\to
R\pi_{*}R\underline{\operatorname{Hom}}(\rho_{*}\widetilde{L},\omega_{X_{B}/B}[1])^{\vee}$

given by the Grothendieck-Serre duality for $X_{B}/B$. Taking this into
account, commutativity of the left square in (10.7) becomes a basic
functoriality of $SD_{\pi}$. On the other hand, commutativity of the right
square in (10.7) similarly follows from the functoriality of
$SD_{\pi\circ\rho}$ applied to the natural morphism $\widetilde{L}\to
Q_{*}{\mathcal{O}}_{S}$ and from the above compatibility for $f=\pi\circ\rho$,
$g=Q$ and $F={\mathcal{O}}_{S}$. ∎

######  Lemma 10.4.

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
${\mathcal{F}}$ 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 $B$, where in
the right vertical arrow we use isomorphisms
${\mathcal{B}}({\mathcal{O}}_{X})\simeq{\mathcal{B}}({\mathcal{O}}_{Y})$ and
${\mathcal{B}}(L)\simeq{\mathcal{B}}(\rho^{*}L)$ coming from the exact
sequences

$0\to{\mathcal{O}}_{X}\to\rho_{*}{\mathcal{O}}_{Y}\to{\mathcal{O}}_{Q}\to 0$
$0\to L\to L\otimes\rho_{*}{\mathcal{O}}_{Y}\to L\otimes{\mathcal{O}}_{Q}\to
0$

and the trivialization of $Q^{*}L$.

###### Proof.

The question is local in the base, so we can assume that we can choose a
relative divisor $D\subset X$ supported in the smooth locus of $\pi:X\to B$,
and a global section $s$ of $L(D)$ such that the $E=\operatorname{div}(s)$ 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

$\displaystyle R\pi_{*}({\mathcal{O}}_{X}):$
$\displaystyle[\pi_{*}L(D)\to\pi_{*}(L(D)|_{E})],$ $\displaystyle
R\pi_{*}(\rho_{*}{\mathcal{F}}):$
$\displaystyle[\pi_{*}(\rho_{*}{\mathcal{F}}\otimes
L(D))\to\pi_{*}(\rho_{*}{\mathcal{F}}\otimes L(D)|_{E})],$ $\displaystyle
R\pi_{*}(L):$ $\displaystyle[\pi_{*}(L(D))\to\pi_{*}(L(D)|_{D})],$
$\displaystyle R\pi_{*}(\rho_{*}{\mathcal{F}}\otimes L):$
$\displaystyle[\pi_{*}(\rho_{*}{\mathcal{F}}\otimes
L(D))\to\pi_{*}(\rho_{*}{\mathcal{F}}\otimes L(D)|_{D})],$

while for the bottom horizontal arrow we use similar resolutions on $Y$ that
use the section $\rho^{*}s$ of $\rho^{*}L(D^{\prime})$, e.g.,

$R\pi_{*}({\mathcal{O}}_{Y}):[\pi^{\prime}_{*}(\rho^{*}L(D^{\prime}))\to\pi^{\prime}_{*}\rho^{*}L(D)|_{E^{\prime}}],$

where $\pi^{\prime}:Y\to B$ is the projection, $D^{\prime}=\rho^{-1}(D)$,
$E^{\prime}=\rho^{-1}(E)$. Note that $D^{\prime}\to D$ and $E^{\prime}\to E$
are isomorphisms, and the assertion follows from the commutativity of the
squares ∎

######  Lemma 10.5.

The following diagram is commutative up to a sign

$\begin{diagram}$ (10.8)

###### Proof.

Note that it is enough to prove the commutativity of this diagram working in
the formal neighborhood $S$ of a stable supercurve $X_{0}$ with one NS node.
We also choose standard coordinates in a formal neighborhood of the node on
$X_{0}$, and use constructions of Sec. 9.3.

Step 1. Under isomorphism (10.6), the trivialization of $N_{B}$ induced by the
equation $t=0$ of the boundary divisor, corresponds to the trivialization of
$\Psi_{1}^{-1}\otimes\Psi_{2}^{-1}$ given by
$\theta_{1}^{-1}\otimes\theta_{2}^{-1}|_{B}$ (where we identify $\Psi_{i}$
with $\pi_{*}{\mathcal{O}}_{D_{i}}/{\mathcal{O}}_{S}$). Below we will use the
notation from Sec. 8.

First, let us consider the canonical section $c$ of
$\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to j_{*}\Omega_{U/S}]$ (where
$j_{*}\Omega_{U/S}$ is placed in degree $0$). The proof of Proposition 8.4(i)
shows that the complex $\pi_{*}[\Omega_{X/S}\rTo{\iota}j_{*}\Omega_{U/S}]$ can
be represented by a morphism of trivial bundles with bases $(b_{i})$,
$(c_{i})$ such that $b_{1},c_{1},b_{4},c_{4}$ are even;
$b_{2},c_{2},b_{3},c_{3}$ are odd; and the differential $\iota$ is given by

$\iota(b_{1})=t^{2}c_{1},\ \ \iota(b_{i})=tc_{i},\text{ for }i=2,3,4;\ \
\iota(b_{i})=c_{i},\ \text{ for }i>4.$

In addition, over $B=S_{0}$, the elements $b_{1},\ldots,b_{4}$ (resp.,
$c_{1},\ldots,c_{4}$) induce the standard bases of the sheaves
$\pi_{*}{\mathcal{K}}$ and $\pi_{*}{\mathcal{C}}_{0}$ from Lemma 8.2. This
shows that the restriction of $c/t$ to $B=S_{0}$ corresponds to the
trivialization of

$\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to
j_{*}\Omega_{U/S}]|_{B}\simeq\operatorname{Ber}\pi_{*}{\mathcal{C}}_{0}\otimes(\operatorname{Ber}\pi_{*}{\mathcal{K}})^{-1}$

induced by the standard bases of $\pi_{*}{\mathcal{K}}$ and
$\pi_{*}{\mathcal{C}}_{0}$.

Note that in the commutative square the bottom horizontal arrow and the right
vertical arrows are injective. Hence, the subsheaves
${\mathcal{K}}\subset\Omega_{X_{B}/B}$ in Lemma 8.2 and in Lemma 10.1(iii) are
the same. This also means that we have an exact sequence

$0\to\rho_{*}\Omega_{\widetilde{X}/B}\to
j_{*}\Omega_{U/S}|_{X_{B}}\to{\mathcal{C}}_{0}\to 0$

and hence, an exact sequence

$0\to{\mathcal{C}}_{0}\to\Omega_{\widetilde{X}/B}(D_{1}+D_{2})|_{D_{1}+D_{2}}\to\widetilde{{\mathcal{C}}}\to
0$ (10.9)

where $\widetilde{{\mathcal{C}}}$ is the cokernel of the right vertical arrow
in the above diagram (see Lemma 10.1(iv)).

Recall that the trivialization of $\pi_{*}{\mathcal{K}}$ used in Lemma
10.1(iii) comes from the filtration of ${\mathcal{K}}$ with the subfactors
$Q_{*}(\Psi_{1}^{2}\Psi_{2}^{2})$, $Q_{*}(\Psi_{1}^{2}\Psi_{2})$,
$Q_{*}(\Psi_{1}\Psi_{2}^{2})$ and $Q_{*}(\Psi_{1}\Psi_{2})$. It is easy to
check that this filtration coincides with the filtration coming from the basis
(8.2) of ${\mathcal{K}}$. Let us consider the following basis of
$\pi_{*}\Omega_{\widetilde{X}/B}(D_{1}+D_{2})|_{D_{1}+D_{2}}$:

$e_{i}:=\frac{dz_{i}-\theta_{i}d\theta_{i}}{z_{i}},\ \
\frac{\theta_{i}dz_{i}}{z_{i}},\ \ \frac{d\theta_{i}}{z_{i}},\ \
\frac{\theta_{i}d\theta_{i}}{z_{i}},\ \ i=1,2.$

Note that it is compatible (up to a sign) with the canonical trivialization of
$\operatorname{Ber}(\pi_{*}\Omega_{\widetilde{X}/B}(D_{1}+D_{2})|_{D_{1}+D_{2}}$.
The filtration of $\widetilde{{\mathcal{C}}}$ considered in Lemma 10.1(iv) is
compatible with this basis: the subsheaf ${\mathcal{O}}_{Q}$ corresponds to
the image of $e_{1}$ (or equivalently of $e_{2}$); the next subfactor
${\mathcal{O}}_{Q}$ is given by the image of $\theta_{1}d\theta_{1}/z_{1}$ (or
of $\theta_{2}d\theta_{2}/z_{2}$); and the quotient
$Q_{*}(\Psi_{1}^{-1}\oplus\Psi_{2}^{-1}$ is given by the image of
$(d\theta_{1}/z_{1},d\theta_{2}/z_{2})$. Hence, the trivialization of
$\operatorname{Ber}\pi_{*}\widetilde{{\mathcal{C}}}$ coming from this basis is
compatible with the isomorphism

$\operatorname{Ber}\pi_{*}\widetilde{{\mathcal{C}}}\simeq(\Psi_{1}^{-1}\Psi_{2}^{-1})^{-1}$

and the trivializations of $\Psi_{i}^{-1}$ given by the image of
$d\theta_{i}/z_{i}$, i.e., by the generator $\theta_{i}$ under the
identification
$\Psi_{i}^{-1}\simeq\pi_{*}{\mathcal{O}}_{D_{i}}/{\mathcal{O}}_{S}$.

On the other hand, the images in ${\mathcal{C}}_{0}$ of the basis vectors

$e_{1}-e_{2},\frac{\theta_{1}d\theta_{1}}{z_{1}}-\frac{\theta_{2}d\theta_{2}}{z_{2}},\frac{\theta_{1}dz_{1}}{z_{1}},\frac{\theta_{2}dz_{2}}{z_{2}}$

are given by $e-f,f,\theta_{1}e,\theta_{2}e$ in terms of the basis (8.3) of
${\mathcal{C}}_{0}$. Hence, the trivialization of
$\operatorname{Ber}\pi_{*}{\mathcal{C}}_{0}$ coming from the latter basis
coincides up to a sign with its trivialization induced by the isomorphism

$\operatorname{Ber}\pi_{*}{\mathcal{C}}_{0}\simeq\operatorname{Ber}^{-1}\pi_{*}\widetilde{{\mathcal{C}}}\simeq\Psi_{1}^{-1}\Psi_{2}^{-1},$

coming from the exact sequence (10.9), and by the trivialization of
$\Psi_{1}^{-1}\Psi_{2}^{-1}$ given by $\theta_{1}\otimes\theta_{2}$.

Combining all the above steps we see that the trivialization $c/t|_{B}$ of
$\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to j_{*}\Omega_{U/S}]|_{B}$ coincides
up to a sign with the trivialization coming from the isomorphism

$\operatorname{Ber}\pi_{*}[\Omega_{X/S}\to
j_{*}\Omega_{U/S}]|_{B}\simeq\Psi_{1}^{-1}\Psi_{2}^{-1}$

and the trivialization $\theta_{1}\otimes\theta_{2}$ of
$\Psi_{1}^{-1}\Psi_{2}^{-1}$.

Step 2. Let

$\phi:{\mathcal{B}}(\omega_{X/\overline{\SS}}^{2})\to{\mathcal{B}}(\omega_{X/\overline{\SS}})^{-2}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-1}(\Delta)$

be isomorphism (9.8). Using the equation $(t=0)$ of $\Delta$, we get an
isomorphism

$t\phi:{\mathcal{B}}(\omega_{X/\overline{\SS}}^{2})\to{\mathcal{B}}(\omega_{X/\overline{\SS}})^{-2}\otimes{\mathcal{B}}({\mathcal{O}}_{X})^{-1}.$
(10.10)

Our goal in this step is to compute it.

We start by recalling the exact sequence (which depends on a choice of
coordinates)

$0\to\omega^{\operatorname{reg}}_{X/S}\to\omega_{X/S}\to{\mathcal{O}}_{Q}\to
0$ (10.11)

(see Lemma 9.3). On the other hand, by Lemma 10.1(ii), we have a natural exact
sequence

$0\to\omega_{\widetilde{X}/B}\to\omega_{X_{B}/B}\to{\mathcal{O}}_{Q}\to 0.$
(10.12)

It is easy to see that the restriction of the embedding
$\omega^{\operatorname{reg}}_{X/S}\to\omega_{X/S}$ to $B$ gives a morphism

$\omega^{\operatorname{reg}}_{X/S}|_{X_{B}}\to\omega_{X_{B}/B}$

with the image $\omega_{\widetilde{X}/B}$, so that the restriction of (10.11)
is compatible with (10.12).

Recall that we choose a sufficiently positive effective divisor $D\subset X$
with support in the smooth locus and a global section $s$ of
$\omega^{\operatorname{reg}}_{X/S}(D)$ as in Lemma 9.3. Let us set

${\mathcal{F}}^{\operatorname{reg}}:=\omega^{\operatorname{reg}}_{X/S}(D)/(s).$

We can modify the derivation of the isomorphism (9.7) by replacing the
resolutions for $R\pi_{*}({\mathcal{O}}_{X})$ with

$[\pi_{*}(\Pi\omega^{\operatorname{reg}}_{X/S}(D))\to\pi_{*}{\mathcal{F}}^{\operatorname{reg}}].$

This leads to an isomorphism

${\mathcal{B}}(\omega_{X/S}^{2})\simeq{\mathcal{B}}({\mathcal{O}}_{X})^{-1}\otimes{\mathcal{B}}(\Pi\omega^{\operatorname{reg}}_{X/S})\otimes{\mathcal{B}}(\Pi\omega_{X/S})\otimes{\mathcal{B}}({\mathcal{G}})\otimes{\mathcal{B}}({\mathcal{F}}^{\operatorname{reg}})^{-1}.$
(10.13)

In addition, exact sequence (10.11) gives an isomorphism

${\mathcal{B}}(\omega^{\operatorname{reg}}_{X/S})\simeq{\mathcal{B}}(\omega_{X/S}).$
(10.14)

On the other hand, we have an isomorphism

$\alpha^{\operatorname{reg}}=(\frac{\mu_{s_{1}}}{t},\alpha_{E}):{\mathcal{F}}^{\operatorname{reg}}\rTo{\sim}{\mathcal{G}}$

induced by some isomorphism $\alpha_{E}$ of the parts supported on $E$ and by
$\mu_{s_{1}}/t$ on the parts supported on $Z$. Namely, near $Z$,
${\mathcal{F}}^{\operatorname{reg}}$ has an ${\mathcal{O}}_{S}$-basis
$s_{1},\theta_{1}s_{1}$ and $\mu_{s_{1}}/t$ sends this basis to the basis
$e,\theta_{1}e$ of ${\mathcal{G}}$.

Hence, we get the induced isomorphism

$\operatorname{ber}(\pi_{*}\alpha^{\operatorname{reg}}):{\mathcal{B}}({\mathcal{F}}^{\operatorname{reg}})\rTo{\sim}{\mathcal{B}}({\mathcal{G}})$
(10.15)

Now we obtain that $t\phi$ is the isomorphism induced by (10.13), together
with (10.14) and (10.15).

Step 3. Let us set

${\mathcal{H}}:=\omega^{2}_{X_{B}/B}/\omega^{2}_{\widetilde{X}/B}.$

Let also $\widetilde{s}$ be the global section of $\omega_{\widetilde{X}}(D)$
induced by $s|_{X_{B}}$, with the zero divisor
$\widetilde{E}\subset\widetilde{X}$ (which is disjoint from the preimage of
the node). Let us set $Z_{B}=Z\cap X_{B}\subset X_{B}$. Note that $Z_{B}$ is
supported on the node and the completition of its ideal is generated by
$z_{1}$, $z_{2}$ and $\theta_{1}+\theta_{2}$, so $Z_{B}$ is smooth of
dimension $0|1$ over $B$. We will construct exact sequences

$0\to\Pi{\mathcal{O}}_{Z_{B}}\to{\mathcal{F}}^{\operatorname{reg}}|_{X_{B}}\to\Pi\omega_{\widetilde{X}/B}(D)|_{\widetilde{E}}\to
0,$ (10.16)
$0\to\omega^{2}_{\widetilde{X}/B}(D)|_{\widetilde{E}}\to{\mathcal{G}}|_{X_{B}}\to{\mathcal{H}}/\Pi{\mathcal{O}}_{Q}\to
0,$ (10.17)
$0\to{\mathcal{H}}\to\omega^{2}_{\widetilde{X}/B}(D_{1})|_{D_{1}}\oplus\omega^{2}_{\widetilde{X}/B}(D_{1})|_{D_{1}}\to{\mathcal{O}}_{Q}\to
0.$ (10.18)

such that the following diagram is commutative up to a sign: where the
vertical arrows come from the exact sequences (10.16) and (10.17), and $\tau$
is the trivialization of ${\mathcal{B}}({\mathcal{H}}/\Pi{\mathcal{O}}_{Q})$
coming from (10.18) and the standard bases
$({\mathbf{b}}_{i},\theta_{i}{\mathbf{b}}_{i})$ of
$\omega^{2}_{\widetilde{X}/B}(D_{i})|_{D_{i}}$.

First, we have a decomposition of ${\mathcal{F}}^{\operatorname{reg}}$ into
the parts supported on $E$ and on $Z$. The latter part is isomorphic to the
completion $\hat{{\mathcal{F}}}^{\operatorname{reg}}$ of
${\mathcal{F}}^{\operatorname{reg}}$. It is easy to see that the section
$s_{1}$ induces an isomorphism

$\Pi{\mathcal{O}}_{Z}\rTo{\sim}\hat{{\mathcal{F}}}^{\operatorname{reg}},$

so we obtain a split exact sequence (10.16).

Next, we have an injective morphism of exact sequences Passing to the
quotients we get (10.17).

Exact sequence (10.18) is immediately obtained from Lemma 10.1. This sequence
shows that ${\mathcal{H}}$ has an ${\mathcal{O}}_{B}$-basis
$(\overline{e},\theta_{1}\overline{e},\theta_{2}\overline{e})$, where
$\overline{e}$ is the image of the local generator $e$ of
$\omega^{2}_{X_{B}/B}$. The embedding $\Pi{\mathcal{O}}_{Q}\to{\mathcal{H}}$
is given by $s_{0}(s_{1}+s_{2})=(\theta_{1}-\theta_{2})\overline{e}$ It
follows that $\overline{e},\theta_{1}\overline{e}$ is a basis of
${\mathcal{H}}/\Pi{\mathcal{O}}_{Q}$. Now we observe that all three
isomorphisms

$\Pi{\mathcal{O}}_{Z_{B}}\rTo{\sim}{\mathcal{F}}^{\operatorname{reg}}_{Z}|_{X_{B}}\rTo{\mu_{s_{1}}/t}{\mathcal{G}}|_{X_{B}}\rTo{\sim}{\mathcal{H}}/\Pi{\mathcal{O}}_{Q}$

send standard ${\mathcal{O}}_{S}$-bases to each other. This gives the desired
commutative diagram.

Step 4. From sequences (10.16), (10.17) and (10.18), together with the
identifications

${\mathcal{B}}(\omega^{2}_{\widetilde{X}/B}(D_{i})|_{D_{i}})\simeq\Psi_{i}^{-1}$

(see Lemma 9.7), we get isomorphisms

${\mathcal{B}}({\mathcal{G}}|_{X_{B}})\otimes{\mathcal{B}}({\mathcal{F}}^{\operatorname{reg}}|_{X_{B}})^{-1}\simeq{\mathcal{B}}({\mathcal{H}})\simeq\Psi_{1}^{-1}\Psi_{2}^{-1}.$
(10.19)

We claim that the top horizontal arrow in diagram (10.8) composed with the
isomorphism ${\mathcal{O}}(\Delta)|_{B}\simeq\Psi_{1}^{-1}\Psi_{2}^{-1}$ gets
identified with the composition of (10.13) with (10.19) (we also take into
account (10.14)).

Indeed, if we use the standard trivialization of
${\mathcal{O}}_{S}(\Delta)|_{B}$, then the top horizontal arrow in (10.8) is
precisely the isomorphism $(t\phi)|_{B}$. By Step 2, it is obtained as the
composition of isomorphisms (10.13) and (10.15), restricted to $B$. By Step 3,
we can replace (10.15) by (10.19) together with the standard trivialization of
$\Psi_{1}^{-1}\Psi_{2}^{-1}$. By Step 1, the latter trivialization corresponds
to the standard trivialization of
${\mathcal{O}}_{S}(\Delta)|_{B}\simeq\Psi_{1}^{-1}\Psi_{2}^{-1}$, so our claim
follows.

Step 5. We see that the restriction of isomorphism (10.13) to $B$ coincides
with the isomorphism

${\mathcal{B}}(\omega_{X_{B}/B}^{2})\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B})^{-1}\otimes{\mathcal{B}}(\Pi\omega_{X_{B}/B})^{-1}\otimes{\mathcal{B}}({\mathcal{O}}_{X_{B}})\to{\mathcal{B}}({\mathcal{G}}|_{X_{B}})\otimes{\mathcal{B}}({\mathcal{F}}^{\operatorname{reg}}|_{X_{B}})^{-1}$
(10.20)

obtained by using resolutions

$\displaystyle R\pi_{*}({\mathcal{O}}_{X_{B}}):$
$\displaystyle[\pi_{*}\Pi\omega^{\operatorname{reg}}_{X/S}(D)|_{X_{B}}\to\pi_{*}{\mathcal{F}}^{\operatorname{reg}}|_{X_{B}}],$
(10.21) $\displaystyle R\pi_{*}(\Pi\omega_{X_{B}/B}):$
$\displaystyle[\pi_{*}\omega_{X_{B}/B}^{2}(D))\to\pi_{*}{\mathcal{G}}|_{X_{B}}],$
(10.22) $\displaystyle R\pi_{*}(\Pi\omega_{\widetilde{X}/B}):$
$\displaystyle[\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)\to\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)|_{D}],$
(10.23) $\displaystyle R\pi_{*}(\omega_{X_{B}/B}^{2}):$
$\displaystyle[\pi_{*}\omega_{X_{B}/B}^{2}(D)\to\pi_{*}\omega_{X_{B}/B}^{2}(D)|_{D}],$
(10.24)

and using the isomorphism
$\operatorname{Ber}(\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)|_{D})\simeq\operatorname{Ber}(\pi_{*}\omega_{X_{B}/B}^{2}(D)|_{D})$.

To prove the commutativity of (10.8) we need to compare (10.20) with the
similar isomorphism where ${\mathcal{O}}_{X_{B}}$ (resp., $\omega_{X_{B}/B}$
and $\omega_{X_{B}/B}^{2}$) is replaced by ${\mathcal{O}}_{\widetilde{X}}$
(resp., $\omega_{\widetilde{X}}$ and $\omega_{\widetilde{X}/B}^{2}$). For this
we will modify resolutions (10.21), (10.23) and (10.24) in a controlled
manner.

We start by noticing that the exact sequence

$0\to{\mathcal{O}}_{X_{B}}\to{\mathcal{O}}_{\widetilde{X}}\to{\mathcal{O}}_{Q}\to
0$

is represented by the exact triangle

$[\Pi{\mathcal{O}}_{Q}\to\Pi{\mathcal{O}}_{Z_{B}}]\to[\Pi\omega^{\operatorname{reg}}_{X/S}(D)|_{X_{B}}\to{\mathcal{F}}^{\operatorname{reg}}|_{X_{B}}]\to[\Pi\omega_{\widetilde{X}/B}(D)\to\Pi\omega_{\widetilde{X}/B}(D)|_{\widetilde{E}}]\to\ldots,$

which measures the difference between resolution (10.21) and the corresponding
resolution of $R\pi_{*}({\mathcal{O}}_{\widetilde{X}})$.

Next, the exact sequence

$0\to\Pi\omega_{\widetilde{X}/B}\to\Pi\omega_{X_{B}/B}\to\Pi{\mathcal{O}}_{Q}\to
0$

is realized by the exact triangle of resolutions

$[\omega^{2}_{\widetilde{X}/B}(D)\to\omega^{2}_{\widetilde{X}/B}(D)|_{\widetilde{E}}]\to[\omega^{2}_{X_{B}/B}(D)\to{\mathcal{G}}|_{X_{B}}]\to[{\mathcal{H}}\to{\mathcal{H}}/\Pi{\mathcal{O}}_{Q}].$

This gives the modification of resolution (10.23).

Finally, the exact sequence

$0\to\omega^{2}_{\widetilde{X}/B}\to\omega^{2}_{X_{B}/B}\to{\mathcal{H}}\to 0$

is realized by the exact triangle of resolutions

$[\omega^{2}_{\widetilde{X}/B}(D)\to\omega^{2}_{\widetilde{X}/B}(D)|_{D}]\to[\omega^{2}_{X_{B}/B}(D)\to\omega^{2}_{X_{B}/B}(D)|_{D}]\to[{\mathcal{H}}\to
0],$

which gives the modification of resolution (10.24).

The modified resolutions,

$\displaystyle R\pi_{*}({\mathcal{O}}_{\widetilde{X}_{B}}):$
$\displaystyle[\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)\to\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)|_{\widetilde{E}}],$
$\displaystyle R\pi_{*}(\Pi\omega_{\widetilde{X}/B}):$
$\displaystyle[\pi_{*}\omega_{\widetilde{X}/B}^{2}(D)\to\pi_{*}\omega_{\widetilde{X}/B}^{2}(D)|_{\widetilde{E}}],$
$\displaystyle R\pi_{*}(\Pi\omega_{\widetilde{X}/B}):$
$\displaystyle[\pi_{*}(\Pi\omega_{\widetilde{X}/B}(D))\to\pi_{*}\Pi\omega_{\widetilde{X}/B}(D)|_{D}],$
$\displaystyle R\pi_{*}(\omega_{\widetilde{X}/B}^{2}):$
$\displaystyle[\pi_{*}\omega_{\widetilde{X}/B}^{2}(D))\to\pi_{*}\omega_{\widetilde{X}/B}^{2}(D)|_{D}],$

give an isomorphism

${\mathcal{B}}(\omega_{\widetilde{X}/B}^{2})\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B})^{-1}\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B})^{-1}\otimes{\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})\rTo{\sim}{\mathcal{B}}(\omega^{2}_{\widetilde{X}/B}(D)|_{\widetilde{E}})\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B}(D)|_{\widetilde{E}})^{-1}$
(10.25)

Note that the composition of this isomorphism with the natural trivialization
of
${\mathcal{B}}(\omega^{2}_{\widetilde{X}/B}(D)|_{\widetilde{E}})\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B}(D)|_{\widetilde{E}})^{-1}$
is precisely the super Mumford isomorphism for $\widetilde{X}/B$,

${\mathcal{B}}(\omega_{\widetilde{X}/B}^{2})\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B})^{-1}\otimes{\mathcal{B}}(\Pi\omega_{\widetilde{X}/B})^{-1}\otimes{\mathcal{B}}({\mathcal{O}}_{\widetilde{X}})\rTo{\sim}{\mathcal{O}}_{B}.$
(10.26)

From the exact triangles connecting the resolutions above we get the following
commutative square of isomorphisms: where the left vertical arrow comes from
the exact sequences mentioned above. By Step 4, the composition of the top
horizontal and right vertical arrows corresponds to the same composition in
diagram (10.8).

Finally, it is easy to check that the following diagram of isomorphisms is
commutative. This allows us to replace the composition of the left vertical
and bottom horizontal arrows in the previous diagram with the same composition
in diagram (10.8), thus, finishing the proof. ∎

###### Proof of Theorem 10.2.

Let us consider the diagram We claim that each square in this diagram is
commutative. Indeed, the left square is commutative by Lemma 10.3, while the
right square is commutative by Lemma 10.5. The commutativity of the middle
square follows from the commutative diagram Here the lower square is
commutative by Lemma 10.4 applied to the morphism $\rho:\widetilde{X}\to
X_{B}$, the line bundle $\omega_{X_{B}/B}^{2}$ and the coherent sheaf
$\omega_{\widetilde{X}/B}$. The commutativity of the upper square can be
checked using the compatibility of the isomorphism (9.5) with the resolution
$[\omega_{X_{B}/B}\to{\mathcal{O}}_{Q}]$ for
$\rho_{*}\omega_{\widetilde{X}/B}$. ∎

###  10.2. Splitting of the Kodaira-Spencer map at an NS boundary component

######  Lemma 10.6.

There is an isomorphism of exact sequences on $B$,

$\begin{diagram}$ (10.27)

where the lower exact sequence is induced by (10.1).

###### Proof.

We have a natural morphism of exact sequences

$\begin{diagram}$ (10.28)

Furthermore, the lower sequence can be identified with

$0\to\rho_{*}\omega^{-2}_{\widetilde{X}}(-D_{1}-D_{2})\to\rho_{*}{\mathcal{A}}_{\widetilde{X},P_{1},P_{2}}\to\pi^{-1}{\mathcal{T}}_{B}\to
0$

Thus, the left vertical arrow in (10.28) can be identified with the natural
map

$\omega_{X/\overline{\SS}}^{-2}\to\rho_{*}\omega^{-2}_{\widetilde{X}}(-D_{1}-D_{2})$

which has cokernel ${\mathcal{O}}_{Q}$ (by (10.1)). On the other hand, the
morphism ${\mathcal{T}}_{\overline{\SS},B}|_{B}\to{\mathcal{T}}_{B}$ has the
kernel ${\mathcal{O}}_{B}$.

Applying the functor $R\pi_{*}$ to (10.28) we immediately derive commutativity
of the right square in (10.27).

Next, we claim that the coboundary morphism

$\ker(\pi^{-1}{\mathcal{T}}_{\overline{\SS},B}|_{B}\to\pi^{-1}{\mathcal{T}}_{B})\to\operatorname{coker}(\omega_{X/\overline{\SS}}^{-2}\to\rho_{*}\omega^{-2}_{\widetilde{X}}(-D_{1}-D_{2}))$

associated with (10.28) gets identified with the natural map
$\pi^{-1}{\mathcal{O}}_{B}\to{\mathcal{O}}_{Q}$. This is a local statement, so
we can use coordinates as in Sec. 7.2. Note that the section $1$ of
$\pi^{-1}{\mathcal{O}}_{B}$ is represented by the vector field $t\partial_{t}$
in $\pi^{-1}{\mathcal{T}}_{\overline{\SS},B}|_{B}$. It lifts to a vector field
$v$ in ${\mathcal{A}}_{X,X_{B}}$ given by

$v(z_{i})=z_{i},\ \ v(\theta_{i})=\frac{1}{2}\theta_{i},\ \ v(t)=t$

(see Lemma 7.5(i)). The restriction of $v$ to $X_{B}$ corresponds to the
vector field

$z_{1}\partial_{z_{1}}+\frac{1}{2}\theta_{1}\partial_{\theta_{1}}+z_{2}\partial_{z_{2}}+\frac{1}{2}\theta_{2}\partial_{\theta_{2}}$

which lives in ${\mathcal{A}}_{\widetilde{X},P_{1},P_{2}}$. The isomorphism

$\omega_{\widetilde{X}/B}^{-2}(-D_{1}-D_{2})\rTo{\sim}{\mathcal{A}}_{\widetilde{X},P_{1},P_{2}}$

is given by

$z_{i}[dz_{i}|d\theta]^{-2}\mapsto
z_{i}\partial_{z_{i}}+\frac{1}{2}(\partial_{\theta_{i}}+\theta_{i}\partial_{z_{i}})(z_{i})\cdot(\partial_{\theta_{i}}+\theta_{i}\partial_{z_{i}})=z_{i}\partial_{z_{i}}+\frac{1}{2}\partial_{\theta_{i}}$

(see the proof of Lemma 2.9), which immediately implies our claim.

From this we deduce that the following square commutes Applying $R\pi_{*}$ we
get the commutativity of the left square in (10.27). ∎

######  Corollary 10.7.

We have a commutative diagram where the horizontal arrows are induced by the
Kodaira-Spencer isomorphisms (see Proposition 9.1), and the right vertical
arrow is given by (10.2).

Combining the above Corollary with Theorem 10.2, we get the statement of
Theorem C.

###  10.3. Ramond boundary components

Now let $\iota:B\to\overline{\SS}$ be a gluing map for the Ramond boundary
component, i.e., one of the maps (8.7), restricted to the locus of smooth
supercurves. Recall that we have a smooth map of relative dimension $0|1$,

$p:B\to\overline{B}$

to a moduli space $\overline{B}$ of smooth supercurves of smaller genus, where
the universal smooth supercurve over $\overline{B}$ is equipped with a pair of
Ramond punctures $R_{1},R_{2}$. Furthermore, $p$ has a structure of the
principal bundle over the group scheme
$\operatorname{Aut}({\mathcal{P}}_{R_{1}})$ which is an extension of
$\operatorname{\mathbb{Z}}/2$ by $\Phi_{1}^{-1}$. Thus, we have a natural
isomorphism

$K_{B}\simeq p^{*}(K_{\overline{B}}\otimes\Phi_{1}).$

On the other hand, we have an exact sequence on the universal stable
supercurve $X_{B}$ over $B$,

$0\to{\mathcal{O}}_{X_{B}}\to{\mathcal{O}}_{\widetilde{X}}\to{\mathcal{O}}_{R_{1}}\to
0,$

where $\widetilde{X}$ is induced by the universal curve $\overline{X}$ over
$\overline{B}$. Taking push-forwards to $B$ and considering the Berezinians,
we get

$\iota^{*}\operatorname{Ber}_{1}\simeq
p^{*}(\operatorname{Ber}^{\overline{B}}_{1}\otimes\Phi_{1}).$

Note that since $\Phi_{1}^{2}$ is canonically trivial, this leads to an
isomorphism

$\iota^{*}\operatorname{Ber}_{1}^{5}\simeq
p^{*}((\operatorname{Ber}^{\overline{B}}_{1})^{5}\otimes\Phi_{1}).$

We can rewrite (9.11) near $B$ as an isomorphism

$K_{\overline{\SS}}(\Delta)\simeq\operatorname{Ber}_{1}^{5}.$

Thus, the restriction to $B$ gives an isomorphism

$K_{B}\simeq\iota^{*}\operatorname{Ber}_{1}^{5}.$

We conjecture that the following diagram is commutative (up to a sign)

##  Appendix A Relative ampleness criterion

Recall that for every superscheme $X$ let
${\mathcal{N}}_{X}\subset{\mathcal{O}}_{X}$ denote the (nilpotent) ideal
locally generated by odd functions, and denote by $X_{{\operatorname{bos}}}$
the usual scheme such that

${\mathcal{O}}_{X_{{\operatorname{bos}}}}={\mathcal{O}}_{X}/{\mathcal{N}}_{X}.$

######  Lemma A.1.

Let $f:X\to S$ be a morphism of superschemes, where $S$ is purely even. Assume
that $f$ factors through a morphism
$\phi:X\to{\mathbb{P}}({\mathcal{E}}^{\vee})$, where
${\mathcal{E}}={\mathcal{E}}^{+}\oplus{\mathcal{E}}^{-}$ is a supervector
bundle on $S$. Then $\phi$ is a closed embedding if and only if
$\phi|_{X_{{\operatorname{bos}}}}$ is a closed embedding (that necessarily
factors through ${\mathbb{P}}(({\mathcal{E}}^{+})^{\vee})$) and the morphism

$f^{*}{\mathcal{E}}^{-}|_{X_{{\operatorname{bos}}}}\to{\mathcal{N}}_{X}/{\mathcal{N}}_{X}^{2}\otimes\phi^{*}{\mathcal{O}}(1)|_{X_{{\operatorname{bos}}}},$
(A.1)

induced by $\phi$, is surjective.

###### Proof.

The “only if” part is clear, so let us prove the “if” part. Let us set
${\mathcal{N}}={\mathcal{N}}_{X}$ for brevity. The underlying map of
topological spaces for $\phi$ is the same as
$\phi_{X_{{\operatorname{bos}}}}:X_{{\operatorname{bos}}}\to{\mathbb{P}}(({\mathcal{E}}^{+})^{\vee})$,
so we just need to show the surjectivity of the homomorphism of sheaves of
rings

$\phi^{\sharp}:{\bigwedge}^{\bullet}_{{\mathcal{O}}_{{\mathbb{P}}(({\mathcal{E}}^{+})^{\vee})}}(p^{*}{\mathcal{E}}^{-}(-1))={\mathcal{O}}_{{\mathbb{P}}({\mathcal{E}}^{\vee})}\to\phi_{*}{\mathcal{O}}_{X},$

where $p:{\mathbb{P}}({\mathcal{E}}^{\vee})\to S$ is the projection. Both
sides have a natural filtration by powers of an ideal: on
$\phi_{*}{\mathcal{O}}_{X}$ we take the filtration
$(\phi_{*}{\mathcal{N}}^{i})$, while on the exterior algebra the filtration
${\bigwedge}^{\geq i}(\cdot)$. Since the above map is compatible with the
surjective map

${\mathcal{O}}_{{\mathbb{P}}(({\mathcal{E}}^{+})^{\vee})}\to\phi_{*}({\mathcal{O}}_{X}/{\mathcal{N}})=\phi_{*}{\mathcal{O}}_{X_{{\operatorname{bos}}}},$

we see that $\phi^{\sharp}$ is compatible with the filtrations. Hence, it is
enough to check surjectivity on the consecutive quotients, which follows from
the surjectivity of the induced map

$p^{*}{\mathcal{E}}^{-}\to\phi_{*}({\mathcal{N}}/{\mathcal{N}}^{2})\otimes{\mathcal{O}}(1)$

or equivalently, of (A.1). ∎

The following criterion generalizes a similar result for supermanifolds in
[21].

Let $f_{{\operatorname{bos}}}:X_{{\operatorname{bos}}}\to
S_{{\operatorname{bos}}}$ be a morphism of usual schemes induced by $f$.

######  Proposition A.2.

Let $f:X\to S$ be a flat morphism of superschemes. If a line bundle $L$ on $X$
is such that $L_{{\operatorname{bos}}}:=L|_{X_{{\operatorname{bos}}}}$ is
strongly relatively ample over $S_{{\operatorname{bos}}}$, then $L$ is
strongly relatively ample over $S$.

###### Proof.

First, let us show that for every coherent sheaf ${\mathcal{F}}$ on $X$, for
$n\gg 0$ one has $R^{>0}f_{*}({\mathcal{F}}\otimes L^{n})=0$ and the natural
map

$f^{*}f_{*}({\mathcal{F}}\otimes L^{n})\to{\mathcal{F}}\otimes L^{n}$ (A.2)

is surjective. Let us consider the commutative diagram We know that the above
assertion is true with $(X,f)$ replaced by
$(X_{{\operatorname{bos}}},f_{{\operatorname{bos}}})$. Since each
${\mathcal{F}}$ has a finite filtration with subsequent quotients that are
scheme-theoretically supported on $X_{{\operatorname{bos}}}$, we can assume
that ${\mathcal{F}}=i_{*}{\mathcal{F}}^{\prime}$ with ${\mathcal{F}}^{\prime}$
a coherent sheaf on $X_{{\operatorname{bos}}}$. We have

$R^{p}f_{*}(i_{*}({\mathcal{F}}^{\prime})\otimes
L^{n})=j_{*}R^{p}f_{{\operatorname{bos}}*}({\mathcal{F}}^{\prime}\otimes
L_{{\operatorname{bos}}}^{n}),$

which immediately implies the vanishing of higher direct images. Furthermore,
the restriction of the map (A.2) to $X_{{\operatorname{bos}}}$ is the map

$i^{*}f^{*}f_{*}i_{*}({\mathcal{F}}^{\prime}\otimes
L_{{\operatorname{bos}}}^{n})=f_{{\operatorname{bos}}}^{*}f_{{\operatorname{bos}}*}({\mathcal{F}}^{\prime}\otimes
L_{{\operatorname{bos}}}^{n})\to({\mathcal{F}}^{\prime}\otimes
L_{{\operatorname{bos}}}^{n})$

which is surjective. Hence, (A.2) is also surjective.

Applying the above statement for ${\mathcal{F}}={\mathcal{O}}_{X}$ we see that
for $n\gg 0$, $R^{>0}f_{*}(L^{n})=0$ and $f^{*}f_{*}(L^{n})\to L^{n}$ is
surjective. In particular, in this case $f_{*}(L^{n})$ is locally free (here
we use flatness of $f$) and the map

$X\to{\mathbb{P}}(f_{*}(L^{n})^{\vee})$ (A.3)

of superschemes over $S$ is well defined. We want to check that (A.3) is a
closed embedding. Let us consider the cartesian square with
$S_{0}:=S_{{\operatorname{bos}}}$ (however, $X_{0}\neq
X_{{\operatorname{bos}}}$ in general). For $n\gg 0$ the above assertions also
hold for $(X_{0},f_{0},L^{n}_{0})$, where $L_{0}=L|_{X_{0}}$, and the base
change map

$f_{*}(L^{n})|_{S_{0}}\to f_{0*}(L_{0}^{n})$

is an isomorphism. Thus, the map $X_{0}\to{\mathbb{P}}(f_{0*}(L^{n})^{\vee})$
is obtained by the base change $S_{0}\to S$ from the map (A.3). Since $S_{0}$
is defined by a nilpotent ideal in $S$, it is enough to check that the map
$X_{0}\to{\mathbb{P}}(f_{0*}(L^{n})^{\vee})$ is a closed embedding for $n\gg
0$.

Thus, we can assume that $S=S_{0}$. Then we have a decomposition of the
supervector bundle ${\mathcal{E}}:=f_{*}(L^{n})$ into even and odd components,

${\mathcal{E}}={\mathcal{E}}^{+}\oplus{\mathcal{E}}^{-}.$

First, we claim that for $n\gg 0$, the map
$X_{{\operatorname{bos}}}\to{\mathbb{P}}(({\mathcal{E}}^{+})^{\vee})$,
corresponding to the surjection

$f^{*}f_{*}(L^{n})|_{X_{{\operatorname{bos}}}}\to
L^{n}|_{X_{{\operatorname{bos}}}},$

is an embedding. Indeed, for $n\gg 0$, we have a surjection

${\mathcal{E}}=f_{*}(L^{n})\to
f_{*}(i_{*}{\mathcal{O}}_{X_{{\operatorname{bos}}}}\otimes
L^{n})=f_{{\operatorname{bos}}*}(L_{{\operatorname{bos}}}^{n}),$

which factors through a surjection ${\mathcal{E}}^{+}\to
f_{{\operatorname{bos}}*}(L_{{\operatorname{bos}}}^{n})$. Since
$L_{{\operatorname{bos}}}$ is relatively ample, we get a composition of two
closed embeddings

$X_{{\operatorname{bos}}}\hookrightarrow\operatorname{Proj}(S^{\bullet}(f_{{\operatorname{bos}}*}(L_{{\operatorname{bos}}}^{n})))\hookrightarrow{\mathbb{P}}(({\mathcal{E}}^{+})^{\vee}),$

and our claim follows.

By Lemma A.1, it remains to show that the map

$f^{*}{\mathcal{E}}^{-}|_{X_{{\operatorname{bos}}}}\to{\mathcal{N}}_{X}\otimes
L^{n}|_{X_{{\operatorname{bos}}}}$

is surjective. We know that for $n\gg 0$, the map

$f^{*}f_{*}({\mathcal{N}}_{X}\otimes L^{n})\to{\mathcal{N}}_{X}\otimes L^{n}$

is surjective. Hence, its restriction to $X_{{\operatorname{bos}}}$ is still
surjective. Since ${\mathcal{N}}_{X}$ is generated by odd functions, we have

${\mathcal{N}}_{X}|_{X_{{\operatorname{bos}}}}=({\mathcal{N}}_{X}|_{X_{{\operatorname{bos}}}})^{-}.$

Hence, we get the surjectivity of the map

$f^{*}(f_{*}({\mathcal{N}}_{X}\otimes
L^{n})^{-})|_{X_{{\operatorname{bos}}}}\to{\mathcal{N}}_{X}\otimes
L^{n}|_{X_{{\operatorname{bos}}}}.$

But $f_{*}({\mathcal{N}}_{X}\otimes L^{n})^{-}$ is a subsheaf in
$f_{*}(L^{n})^{-}={\mathcal{E}}^{-}$, so we are done. ∎

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