# On super curves and supervolumes

Ricardo Jesús Ramos Castillo Email<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

We study the geometry of super curves with a chosen supervolume form. We
consider the algebra of divergence free vector fields $S(1|N)$ associated to
such curves. When $N=2$ its derived algebra, called $S(2)$, defines a special
family of curves, named $S(2)$-super curves. We exhibit an involution on the
moduli space of such curves that generalizes Deligne’s involution for $N=1$
super curves. The fixed point set of this involution consists on Manin’s
$SUSY_{2}$-super curves. We describe the moduli spaces of these curves.

##  1 Introduction

######  1.1.

In his famous work on classification of Lie super algebras [1], Kac introduced
a list of infinite dimensional Lie super algebras generalizing the ordinary
theory of Lie algebras of Cartan type. These are subalgebras of derivations on
a super-commutative algebra freely generated by $n$ even variables and $N$ odd
variables. In the case $n=1$ these are algebras of vector fields on a
punctured super-disc
$\mathrm{Spec}\mathbb{C}((t))[\theta_{1},\dots,\theta_{N}]$ preserving some
extra structure. In this case, the list consists of

  1. 1.

$W(1|N)$, all vector fields.

  2. 2.

$S(1|N)$, divergence free vector fields, that is, vector fields acting
trivially on the Berezininan, or preserving the section $[dt|d\theta_{1}\dots
d\theta_{N}]$ of the Berezinian bundle.

  3. 3.

$CS(1|N)$, vector fields that preserve the Berezinian up to multiplication by
a scalar function.

  4. 4.

$K(1|N)$, vector fields preserving a _contact-like_ form

$dt+\sum\theta_{i}d\theta_{i},$

up to multiplication by a scalar function.

Some of these algebras are isomorphic, for example $W(1|1)\simeq K(1|2)$.
While some others are not simple, for example $S(1|2)$ is not simple, but its
derived algebra $S(2)$ is.

For each such Lie super algebra, there is an associated class of super curves,
with certain geometric structures preserved by these vector fields. That is,
the class of super curves admitting an etále cover where infinitesimal changes
of coordinates are given by vector fields in the corresponding Lie super
algebra. For example, a $W(1|1)$-super curve is a general $1|1$-dimensional
super curve, they consist of a smooth curve $C$ together with a line bundle
$\mathcal{L}$ over it. $K(1|1)$-super curves are called $SUSY$-super curves by
Manin in [2] and $SUSY$-super Riemman surfaces in [3, 4], they consist of a
smooth curve $C$ and a choice of a square root of the canonical bundle
$\omega_{C}$. Similarly $K(1|2)$-super curves are called (oriented)
$SUSY_{2}$-super curves by Manin. In [5] Vaintrob studied the geometry of all
these super curves, obtaining a description of the corresponding moduli spaces
in each case.

In this article we focus on one example that is missing in Vaintrob’s list,
these are the so-called $S(2)$-super curves. These are smooth
$1|2$-dimensional super curves, endowed with a trivializing section of its
Berezinian bundle and with the additional condition that the above mentioned
changes of coordinates lie in Kac’s $S(2)$-algebra as opposed to the full
algebra $S(1|2)$.

######  1.2.

Deligne exploited the isomorphism $W(1|1)\simeq K(1|2)$ in [6] to describe an
involution in the moduli space of general smooth $1|1$-dimensional super
curves, the fixed locus of which is the moduli space of $K(1|1)$-super curves.
This involution is induced by an involution of the Lie super algebra $K(1|2)$,
fixing its subalgebra $K(1|1)$. Geometrically, a $W(1|1)$-super curve, or a
general $1|1$-dimensional super curve, over a purely even super scheme $S$
(that is simply a scheme $S$) is given by a smooth curve $C$ over $S$ together
with a line bundle $\mathcal{L}$ over it. Deligne’s involution corresponds to
taking the Serre dual of $\mathcal{L}$:

$(C,\mathcal{L})\leftrightarrow(C,\omega_{C/S}\otimes_{\mathcal{O}_{C}}\mathcal{L}^{*}).$

The fixed point set of this involution is parametrized by curves $C$ together
with a choice of a _theta-characteristic_ (a square root of the canonical
bundle). This is the moduli space of $K(1|1)$-super curves as shown in [5].

In [7], Donagi and Witten show that when the base $S$ is an arbitrary (non-
necessarily even) super scheme there exist non-split $W(1|1)$ super curves
over $S$. In particular these curves are not given as the spectrum of the free
super commutative algebra generated by a line bundle $\mathcal{L}$ as above.
The description of Deligne’s involution in this case is not so transparent.

######  1.3.

Our main result is to generalize Deligne’s involution to the case of $S(2)$
curves. In order to do so we consider the sequence of inclusions

$K(1|2)\subset S(2)\subset S(1|2)\subset K(1|4).$ (1.1)

This provides a sequence of embeddings of the corresponding moduli spaces:
each $K(1|2)$-super curve comes with a trivialization of its Berezinian bundle
and the local changes of coordinates are in $S(2)$. Each $S(2)$-super curve is
in particular a $1|2$-dimensional super curve with a trivialization of its
Berezinian bundle $\mathrm{Ber}_{C}$ (an $S(1|2)$-super curve). Similarly if
$C$ is a $1|2$-dimensional super curve with a trivialization of
$\mathrm{Ber}_{C}$, consider its tangent bundle $\mathcal{T}_{C}$, a locally
free $\mathcal{O}_{C}$ module of rank $1|2$. The Grassmanian $\widetilde{C}$
of rank $0|2$ subbundles of $\mathcal{T}_{C}$ is a $1|4$-dimensional super
curve with a canonical $K(1|4)$-structure [8]. We show that there exists an
involution of $K(1|4)$ that fixes pointwise its subalgebra $K(1|2)$ and
preserves (but does not fix) $S(2)$. This involution implies

######  Theorem 1.4 (Theorem 4.3).

There exists an involution $\mu$ of the moduli space $\mathcal{M}_{S(2)}$ of
$S(2)$-super curves. The fixed point set of $\mu$ consists of the moduli space
$\mathcal{M}_{K(1|2)}$ of orientable $SUSY_{2}$-super curves.

There are super curves with trivial Berezinian that are not $S(2)$-super
curves. And the Lie super algebra $S(1|2)$ is not stable under the involution
$\mu$ above. This shows that our generalization of Deligne’s involution
requires precisely the $S(2)$-structure as opposed to $S(1|2)$.

######  1.5.

$S(1|2)$ curves admit a simple geometrical description: these are
$1|2$-dimensional super-curves together with a trivialization of its
Berezinian bundle. In contrast, $S(2)$ curves do not admit this simple
geometrical description. To any $S(1|2)$ curve $C$ we attach an affine bundle,
or a $\mathbb{G}_{a}$ torsor $K_{C}$. The class of this bundle is an
obstruction for the $S(1|2)$ curve to be an $S(2)$ curve, namely $C$ is an
$S(2)$ curve if and only if $K_{C}$ is trivial (Proposition 3.15).

######  1.6.

Given a $W(1|1)$-super curve $C$ over a purely even super scheme $S$, it is
split in the sense that there exits a smooth $1|0$-dimensional curve $C_{0}$
over $S$ and a line bundle $\mathcal{L}$ over $C_{0}$ such that
$C=\mathrm{Spec}\operatorname{Sym}_{\mathcal{O}_{C_{0}}}\mathcal{L}[-1]$. This
allowed us to describe Deligne’s involution as taking Serre’s dual. A similar
situation arises in the $S(2)$ case: for a purely even scheme $S$ and an
$S(2)$-super curve $C$ over $S$, there exists a smooth $1|0$-dimensional curve
$C_{0}$ over $S$, a rank two bundle $\mathcal{E}$ over $C_{0}$ satisfying
$\det\mathcal{E}\simeq\omega_{C_{0}/S}$ and such that
$C\simeq\mathrm{Spec}\operatorname{Sym}_{\mathcal{O}_{C_{0}}}\mathcal{E}[-1]$.
That is we have

######  Theorem 1.7 (Theorem 3.17).

Every $S(2)$-super curve over a purely even base $S$ is split.

In this situation, our involution above is given just as in the $W(1|1)$ case:
it corresponds to

$(C_{0},\mathcal{E})\leftrightarrow(C_{0},\omega_{C_{0}/S}\otimes_{\mathcal{O}_{C_{0}}}\mathcal{E}^{*}).$

######  Observation 1.8.

Theorem 1.7 is false for $S(1|2)$ super curves. Even over a purely even base
$S$, there are $S(1|2)$-super curves that are not split (see Example 5.2), and
therefore they are not $S(2)$-super curves.

The condition on the base $S$ on Theorem 1.7 is necessary, that is, there
exists families of $S(2)$-curves over super schemes that are not split (see
Example 5.1).

From this point of view, the condition on a super curve with trivial
Berezinian, of being an $S(2)$-super curve is the analog of the condition of
being oriented for general $SUSY_{2}$-super curves as in [2].

######  1.9.

Our second result is a description of the moduli spaces of $S(2)$ and $S(1|2)$
supercurves. We first characterize the universal family of such curves over a
purely even base $S$:

######  Proposition.

(See Proposition 6.4 below) The data of a family of $S(2)$-supercurves
$C\rightarrow S$ whose reduction coincides with a given family of curves
$C_{0}\rightarrow S$ over a purely even scheme $S$ is equivalent to a rank $2$
vector bundle $E\rightarrow C_{0}$ together with an isomorphism $\det
E\xrightarrow{\beta}\Omega_{C_{0}/S}$. Two such supercurves $(E,\beta)$ and
$(E^{\prime},\beta^{\prime})$ are equivalent if and only if there exists a
bundle isomorphism $\alpha:E\rightarrow E^{\prime}$ such that
$\beta^{\prime}\circ\det\alpha=\beta$.

Similarly for $S(1|2)$ supercurves we have Proposition 6.12:

######  Proposition.

A family of $S(1|2)$ curves $C\rightarrow S$ with a given reduction
$\pi:C_{0}\rightarrow S$ over a purely even base $S$ is determined by a rank
$2$ bundle $E\rightarrow C_{0}$ together with an isomorphism $\beta:\det
E\rightarrow\Omega_{C_{0}/S}$ and a class $\Gamma\in
H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$. Two such supercurves $(E,\beta,\Gamma)$
and $(E^{\prime},\beta^{\prime},\Gamma^{\prime})$ are equivalent if
$\Gamma=\Gamma^{\prime}$ and the pairs $(E,\beta)$ and
$(E^{\prime},\beta^{\prime})$ are equivalent as in the previous proposition.

The map $(E,\beta,\Gamma)\rightarrow(E,\beta)$ could be thought of as a
fibration from the moduli space of $S(1|2)$ supercurves to the moduli space of
$S(2)$ curves over purely even bases. However, there are non-trivial odd
deformations of $S(2)$ supercurves.

We describe the full moduli space of $S(2)$ supercurves under the assumption
that the base super-scheme is split. Given such a split super-scheme $S$ with
purely even reduction $S_{\mathrm{rd}}$. The datum of a family of $S(2)$
curves $C\rightarrow S$ with reduction $C_{0}\rightarrow S_{\mathrm{rd}}$ is
equivalent to a class in $H^{1}(C_{0},G)$ where $G$ is a sheaf of groups over
$C_{0}$ described in 6.3 (see Proposition 6.7).

######  1.10.

The organization of this article is as follows. In section 2 we recall the
basic preliminaries on super-commutative algebras, their modules, and their
derivations. We introduce the relevant infinite dimensional Lie algebras in
Kac’s list and describe their associated infinite dimensional groups as groups
of automorphisms of a super-disk preserving certain geometric structure.

In section 3 we recall the basic preliminaries on super-geometry. Define our
curves of interest and construct a principal $\mathbb{G}_{a}$ bundle
characterizing the obstruction of an $1|2$ dimensional super-curve with
trivial Berezinian being an $S(2)$ super-curve. We show that every $S(2)$
curve over a purely even base is split (Theorem 3.17) and we finish that
section attaching a SUSY $1|2N$ curve to any $1|N$ super-curve.

In section 4 we describe the involution of the moduli space of $S(2)$
supercurves generalizing that of Deligne for general $1|1$ supercurves. In
section 5 we give the above mentioned examples of families of supercurves:
$S(1|2)$ supercurves that are not $S(2)$ supercurves. Non-split $S(1|2)$
supercurves over a purely even base. Non-split $S(2)$ supercurves over a
super-scheme. In section 6 we give a description of the moduli spaces of
$S(1|2)$ and $S(2)$ supercurves. We describe the subspace of supercurves over
purely even schemes and then describe the possible deformations of such a
curve in the odd directions of the base under the assumption that the base is
a split superscheme.

We identify the full automorphism group of such families of supercurves for
genus $g\geq 2$ and describe the corresponding orbifold quotient.

##  2 Super algebra

###  2.1 Super algebras

######  Definition 2.1.

A _$k$ -super algebra $R$ over a field $k$_ is a
$\mathbb{Z}/2\mathbb{Z}$-graded $k$-vector space, $R=R_{0}\oplus R_{1}$111For
super spaces we will write $R_{i}$ meaning the $i$-th part of $R$ for
$i=0,1\mod 2$. with a unital multiplication $R\otimes_{k}R\to R$ that respects
the gradation, i.e. $R_{i}R_{j}\subset R_{i+j}$, such that for homogeneous
elements $a\in R_{i}$, $b\in R_{j}$ we have the commutative rule:
$ba=(-1)^{ij}ab$.

Let $R,S$ be $k$-super algebras, a $k$-linear map $T:R\to S$ is said to be
_even_ if $T(R_{i})\subset S_{i}$, $i=0,1\mod 2$, and is said to be _odd_ if
$T(R_{i})\subset S_{i+1}$, $i=0,1\mod 2$. An even $k$-linear map $T:R\to S$ is
said to be a _homomorphism of super algebras_ if
$T(rr^{\prime})=T(r)T(r^{\prime})$ for any $r,r^{\prime}\in R$ and $T(1)=1$.
The set of super algebras homomorphisms is going to be denoted by
$Hom_{SAlg}(R,S)$. Let $R,S$ be super algebras over $k$, we say that _$R$ is
an $S$ algebra_ if there exists a homomorphism of super algebras $S\to R$.

An element $a\in R$ is called _even_ if $a\in R_{0}$ and is said to be _odd_
if $a\in R_{1}$. Also, we say that $a\in R$ has _parity $j$_ if $a\in R_{j}$.

For a non-nilpotent even element $f\in R_{0}$ we denote by $R_{f}$ the super
algebra given by the localization of $R$ with respect to the multiplicative
set $\\{1,f,f^{2},\dots\\}$.

######  Example 2.2.

A commutative ring $R$ over $k$ can be seen as a super algebra with $R_{0}=R$
and $R_{1}=0$. Also, observe that for any super algebra $R$, $R_{0}$ is a
commutative ring.

######  Example 2.3.

Given a super algebra $R$, the super algebra of polynomials $R[t]$, with $t$
an even variable, is defined as the usual algebra of polynomials with the
$\mathbb{Z}/2\mathbb{Z}$-gradation:

$R[t]_{j}:=\\{a_{0}+a_{1}t+\dots a_{n}t^{n}:n\in\mathbb{N},a_{i}\in R_{j}\\},$

then, $R[t]$ is a super algebra. Recursively, we will consider the super
algebra $R[t_{1},\dots,t_{n}]:=R[t_{1},\dots,t_{n-1}][t_{n}]$, for the even
variables $t_{1},\dots,t_{n}$.

For a super algebra $R$ we can construct the _Grassmann algebra_ $R[\theta]$,
with $\theta$ an odd variable, defined as the usual algebra of polynomials
with the $\mathbb{Z}/2\mathbb{Z}$-gradation:

$R[\theta]_{j}:=\\{a_{0}+a_{1}\theta:a_{0}\in R_{j},a_{1}\in R_{j+1}\\},$

then, $R[\theta]$ is a super algebra with $\theta\in R[\theta]_{1}$.
Recursively, we will consider the Grassmann algebra of rank $n$,
$R[\theta^{1},\dots,\theta^{n}]:=R[\theta^{1},\dots,\theta^{n-1}][\theta^{n}]$,
for the odd variables $\theta^{1},\dots,\theta^{n}$.

We write $R[m|n]:=R[t_{1},\dots,t_{m}][\theta^{1},\dots,\theta^{n}]$, where
$t_{1},\dots,t_{m}$ are even and $\theta^{1},\dots,\theta^{n}$ are odd
variables.

Similarly, we define
$R[[m|n]]:=R[[t_{1},\dots,t_{m}]][\theta^{1},\dots,\theta^{n}]$, for the even
variables $t_{1},\dots,t_{m}$ and odd variables $\theta^{1},\dots,\theta^{n}$.

######  Definition 2.4.

Let $R$ be a super algebra and put $J:=R_{1}+R_{1}^{2}$. We define the
_reduced ring of $R$_ as the quotient $R_{\text{rd}}:=\frac{R}{J}$, this is a
ring endowed with the projection $R\to R_{\text{rd}}$.

######  Observation 2.5.

Let us consider the category of super algebras over a field $k$, $SSch/k$.
Over it, any super algebra $R$ define a functor

$\begin{split}h_{R}:SAlg\to&Sets\\\ S\mapsto&Hom_{SAlg}(R,S).\end{split}$

If we consider a commutative $k$-algebra $S$, then any morphism $R\to S$
vanishes in $R_{1}$, since $S_{1}=0$. Then such morphism factorizes through
the projection $R\to R_{\text{rd}}$, so we get the natural bijection

$Hom_{SAlg}(R,S)\to Hom_{CAlg}(R_{\text{rd}},S),$

where $CAlg$ is the category of commutative $k$-algebras. Finally, we get the
lemma:

######  Lemma 2.6.

The restriction $h_{R}|_{CAlg}$ is represented by $R_{\text{rd}}$.

######  Example 2.7.

For a commutative algebra $R$ and the Grassmann algebra
$R[\theta^{1},\dots,\theta^{n}]$, observe that

$R[\theta^{1},\dots,\theta^{n}]\to(R[\theta^{1},\dots,\theta^{n}])_{\text{rd}}\simeq
R,$

and there exists a section $R\to R[\theta^{1},\dots,\theta^{n}]$.

###  2.2 Super modules

######  Definition 2.8.

Let $R$ be a $k$-super algebra and consider $M$ be
$\mathbb{Z}/2\mathbb{Z}$-graded $k$-vector space, we will say that $M$ is a
_module over $R$_ if is endowed with a $k$-bilinear product

$R\otimes_{k}M\to M$

such that $R_{i}M_{j}\subset M_{i+j}$, and for any $a,b\in R$ and $m\in M$:

$\begin{split}a\cdot(b\cdot m)=&\ (ab)\cdot m,\\\ 1\cdot m=&\ m.\end{split}$

For a super module $M$ we can construct the super module $\Pi M$ by

$\Pi M_{i}:=M_{i+1}.$

######  Example 2.9.

For a super algebra $R$ the super algebra $R[n|m]$ is a module over $R$. In
particular, $R$ is a module over $R$.

The direct sum $R^{m|n}:=R^{m}\oplus(\Pi R)^{n}$ is also a module over $R$.

######  Definition 2.10.

For two super modules $M_{1}$, $M_{2}$ over $R$ and a $k$-linear map
$T:M_{1}\to M_{2}$, we say that $T$ is _even_ if $T(M_{1,i})\subset M_{2,i}$,
$i=0,1$, and _odd_ if $T(M_{1,i})\subset M_{2,i+1}$, $i=0,1$. We say that $T$
is _an homogeneous $R$-homomorphism of $R$ modules_ if $T$ has parity $j$ and
for any $a\in R_{i}$ we have $T(am)=(-1)^{ij}aT(m)$.

The space of $R$-homomorphism of $R$ modules, denoted by
$\underline{\mathrm{Hom}}_{R}(M_{1},M_{2})$, is the
$\mathbb{Z}/2\mathbb{Z}$-graded space of even and odd $R$-homomorphism:

$\underline{\mathrm{Hom}}_{R}(M_{1},M_{2}):=\mathrm{Hom}_{R}(M_{1},M_{2})_{0}\oplus\mathrm{Hom}_{R}(M_{1},M_{2})_{1}.$

We say that $T\in\underline{\mathrm{Hom}}_{R}(M_{1},M_{2})$ is _invertible_ if
there exists an $S\in\underline{\mathrm{Hom}}_{R}(M_{2},M_{1})$ such that
$T\circ S=\mathrm{id}_{M_{2}}$ and $S\circ T=\mathrm{id}_{M_{1}}$. An even
element $T\in\underline{\mathrm{Hom}}_{R}(M_{1},M_{2})$ that is invertible is
called _isomorphism_. In this case, we say that $T$ has _inverse_ $S$ and that
$M_{1}$ and $M_{2}$ are _isomorphic_. When $M_{1}=M_{2}$, an isomorphism
$T:M_{1}\to M_{1}$ is called _automorphism_ instead of isomorphism.

An $R$-module $M$ is _free, finitely generated and has rank $m|n$_ if $M$ is
isomorphic to $R^{m|n}$.

######  Example 2.11.

Let $R$ be a super algebra, for a super module $M$, an element $a\in R_{i}$
induces an $R$-homomorphism with parity $i$ by multiplication:

$\begin{split}T_{a}:M\to&\ M\\\ m\mapsto&\ am.\end{split}$

Observe that any invertible element in $R_{0}$ induce an automorphism in $M$.

From the odd $k$-linear map $M\to\Pi M$, $m\to m$, we obtain $M=\Pi\Pi M$.
Note, that $M\to\Pi M$ is bijective but not an isomorphism since this morphism
is odd.

######  Observation 2.12.

Let $M$ be a super module, and take

$\mathrm{End}_{R}(M):=\underline{\mathrm{Hom}}_{R}(M,M),$

that is a $\mathbb{Z}/2\mathbb{Z}$-graded algebra with the composition as
product. The set of invertible automorphisms is going to be denoted by
$\mathrm{End}_{R}(M)^{*}$.

######  Observation 2.13.

Let $M$ be a free super module of rank $m|n$, i.e. $M$ is isomorphic to
$R^{m}\oplus(\Pi R)^{n}$, observe that

$M_{\text{rd}}:=R_{\text{rd}}\otimes_{R}M$

is a rank $m|n$ free module over $R_{\text{rd}}$.

Observe that if $M$ has rank $m|n$, then $\Pi M$ has rank $n|m$.

######  Example 2.14.

For a finitely generated free super module $M$ of rank $m|n$ over $R$ with
generators $\\{t_{1},\dots,t_{m}|\theta^{1},\dots,\theta^{n}\\}$, with $t_{i}$
even and $\theta^{j}$ odd, the construction given in 2.3 gives us a super
algebra, $S_{R}[M]:=R[t_{1},\dots,t_{m}|\theta^{1},\dots,\theta^{n}]$.
Observe, that this super algebra does not depend on the choice of generators.

######  Definition 2.15.

Let $M$ be a free super module of rank $m|n$ and $T:M\to M$ be an invertible
morphism represented in some basis by the matrix

$T=\left[\begin{matrix}A&B\\\ C&D\end{matrix}\right]$

where $A,B,C,D$ is a $m\times m$, $m\times n$, $n\times m$ and $n\times n$
matrix, respectively, we define the _Berezinian of $T$_ by

$\mathrm{Ber}(T)=\det(A-BD^{-1}C)\det(D)^{-1}.$ (2.1)

######  Observation 2.16.

The Berezinian verifies the following conditions:

  1. 1.

If $T=\left[\begin{matrix}A&B\\\ 0&D\end{matrix}\right]$ or
$T=\left[\begin{matrix}A&0\\\ C&D\end{matrix}\right]$, then
$\mathrm{Ber}(T)=\det(A)\det(D)^{-1}$.

  2. 2.

Let $T,S$ be two automorphisms, then
$\mathrm{Ber}(TS)=\mathrm{Ber}(T)\mathrm{Ber}(S)$. In particular,
$\mathrm{Ber}(T)$ does not depend on the basis chosen.

  3. 3.

Suppose that $k=\mathbb{C}$ and that $M$ is finitely generated, so for
$T\in\mathrm{End}_{k}(M)$ we can define $\exp(T)=\sum_{i\geq
0}\frac{T^{i}}{i!}$. In this case we have

$\mathrm{Ber}(\exp(T))=\exp(\text{str}(T)),$

where $\text{str}(T):=\text{tr}(A)-\text{tr}(D)$ is the _super-trace_ of $T$.

  4. 4.

To define the Berezinian, we just need that $D$ in (2.1) is invertible, and
the observations above still hold even when $T$ is not necessarily invertible.

######  Observation 2.17.

For the free super module $M$ with generators
$\\{t_{1},\dots,t_{n},\theta^{1},\dots,\theta^{m}\\}$, with $t_{i}$ even and
$\theta^{j}$ odd, we can construct the free module $\mathrm{Ber}(M)$ generated
by the formal element $[t_{1},\dots,t_{n}|\theta^{1},\dots,\theta^{m}]$ with
parity $m\mod 2$. Then $\mathrm{Ber}(M)$ has rank $1|0$ if $m$ is even and
rank $0|1$ if $m$ is odd. An invertible homomorphism $T:M\to M$, induce the
automorphism $\mathrm{Ber}(T):\mathrm{Ber}(M)\to\mathrm{Ber}(M)$. Finally, we
get an homomorphism of super groups:

$\begin{split}End_{R}(M)^{*}\to&\ End_{R}(\mathrm{Ber}(M))^{*}\\\ T\mapsto&\
\mathrm{Ber}(T).\end{split}$

###  2.3 Super derivations

######  Definition 2.18.

Let $R$ be a $S$-super algebra and let $D\in End_{S}(R)$ with parity $i$. We
say that $D$ is _an $S$-derivation_ if $D(s)=0$ for any $s\in S$, and for any
$a\in R_{j}$ and $b\in R$ we have:

$D(ab)=D(a)b+(-1)^{ij}aD(b).$

The vector space of derivations has a structure of $R$ super module given by
$(aD)(b)=aD(b),\text{ for any }a,b\in R$. We denote by $\mathrm{Der}_{R/S}$
the _super module of derivations_.

For two derivations $D_{1}\in\mathrm{Der}_{R/S,i}$ and
$D_{2}\in\mathrm{Der}_{R/S,j}$ we define _the bracket_ by:

$[D_{1},D_{2}]=D_{1}D_{2}-(-1)^{ij}D_{2}D_{1},$

with this structure $\mathrm{Der}_{R/S}$ is a super Lie algebra.

######  Example 2.19.

Let $R[m|n]:=R[t_{1},\dots,t_{m}][\theta^{1},\dots,\theta^{n}]$, where
$t_{1},\dots,t_{m}$ are even and $\theta^{1},\dots,\theta^{n}$ are odd, be the
super algebra of polynomials associated to the super algebra $R$, then the set
of derivations of $R[m|n]$ over $R$ is a $R[m|n]$ free module with even part
generated by $\\{\partial_{t_{1}},\dots,\partial_{t_{m}}\\}$ and odd part
generated by $\\{\partial_{\theta^{1}},\dots,\partial_{\theta^{n}}\\}$.

Similarly, the generators of $\mathrm{Der}_{R}(R[[1|n]])$ are given by
$\\{\partial_{t_{1}},\dots,\partial_{t_{m}}|\partial_{\theta^{1}},\dots,\partial_{\theta^{n}}\\}$.

For a vector field
$X=A_{0}\partial_{z}+A_{1}\partial_{\theta^{1}}+\cdots+A_{N}\partial_{\theta^{N}}\in\mathrm{Der}_{R}(R[[1|n]])$,
we define _the super divergence operator_ :

$\mathrm{sdiv}(X):=\partial_{z}A_{0}+(-1)^{A_{1}}\partial_{\theta^{1}}A_{1}+\cdots+(-1)^{A_{N}}\partial_{\theta^{N}}A_{N},$

and denote by $S(1|N)$ the _space of divergence free vector fields_.

######  Observation 2.20.

Let $R[[1|2]]$ be a super algebra with generators
$\\{t|\theta^{1},\theta^{2}\\}$ and maximal ideal $\mathfrak{m}=\langle
t|\theta^{1},\theta^{2}\rangle$. The space of divergence free vector fields in
$\mathrm{Der}_{R}(R[[1|2]]_{\mathfrak{m}})$ has even generators:

$\begin{split}L_{m}=&-z^{m+1}\partial_{z}-\frac{m+1}{2}z^{m}\sum_{i=1}^{N}\theta^{i}\partial_{\theta^{i}},\text{
for $m\in\mathbb{Z}$},\\\
J_{m}^{0}=&z^{m}(\theta^{1}\partial_{\theta^{1}}-\theta^{2}\partial_{\theta^{2}}),\text{
for $m\in\mathbb{Z}$},\\\
J_{m}^{1}=&z^{m}\theta^{1}\partial_{\theta^{2}},\text{ for
$m\in\mathbb{Z}$},\\\ J_{m}^{2}=&z^{m}\theta^{2}\partial_{\theta^{1}},\text{
for $m\in\mathbb{Z}$},\\\ K=&\theta^{1}\theta^{2}\partial_{z},\end{split}$

and odd part with generators:

$\begin{split}G_{m}^{i}=&-z^{m+1/2}\partial_{\theta^{1}},\text{ for
$m\in\mathbb{Z}$},\\\
H_{m}^{i}=&z^{m+1/2}\theta^{i}\partial_{z}-\left(m+1/2\right)z^{m-1/2}\theta^{i}\sum_{j=1}^{N}\theta^{j}\partial_{\theta^{j}},\text{
for $m\in\mathbb{Z}$},.\end{split}$

Then, the algebra $S(1|2)$ is not simple, but its derived algebra

$S(2):=[S(1|2),S(1|2)]$

is simple (this is shown in [9]). Observe that $S(1|2)/S(2)$ is a rank 1 Lie
algebra. The map $S(1|2)\to S(1|2)/S(2)$ has its image generated by
$K=\theta^{1}\theta^{2}\partial_{z}$. For $X\in S(1|2)$, then $X\in S(2)$ if
and only if $\partial_{\theta^{1}}\partial_{\theta^{2}}(X\cdot z)=0$.

###  2.4 The group $\mathrm{Aut}_{R}(R[[m|n]])$

Let $R$ be a super algebra, for the super algebra $R[[m|n]]$ we will consider
the collection of automorphisms of $R$-super algebras and denote this group as
$\mathrm{Aut}_{R}(R[[m|n]])$, that is the set of even maps $T:R[[m|n]]\to
R[[m|n]]$ such that $T|_{R}=id_{R}$. If there is no confusion, we will write
$\mathrm{Aut}(R[[m|n]])$ or $\mathrm{Aut}[[m|n]]$.

We are interested in the group of automorphism $\mathrm{Aut}(R[[1|n]])$ and
its group structure given by $\Phi*\Psi=\Psi\circ\Phi$.

######  Example 2.21.

Let $R$ be a $k$ super algebra, $\mathrm{char}(k)=0$, for a nilpotent
$X\in\mathrm{Der}_{R}(R[[1|n]])_{0}$, we define its exponential by:

$\exp(X)=\mathrm{id}+\frac{X}{1!}+\frac{X^{2}}{2!}+\cdots.$ (2.2)

For two nilpotent $N,L\in\mathrm{Der}_{R}(R[[1|n]])_{0}$ with $[N,L]=0$ we get

$\exp(N+L)=\exp(N)\exp(L).$

In particular, $\exp(N)$ has an inverse $\exp(-N)$. Finally,
$\exp(X)\in\mathrm{Aut}_{R}(R[[m|n]])$.

Let $m\in\mathbb{N}$, and the $R$-super algebra
$R[t|\theta^{1},\cdots,\theta^{n}]/\mathfrak{m}^{m}$, where
$\mathfrak{m}:=\langle t|\theta^{1},\dots,\theta^{n}\rangle$. Similar to
Example 2.21, an element
$X\in\mathrm{Der}_{R}(R[t|\theta^{1},\cdots,\theta^{n}]/\mathfrak{m}^{m})$ is
nilpotent, so we can define the exponential as (2.2). For the ind-family of
$R$-algebras
$\\{R[t|\theta^{1},\cdots,\theta^{n}]/\mathfrak{m}^{m}\\}_{m\in\mathbb{N}}$,
we get the prounipotent group and its pronilpotent Lie algebra:

$\begin{split}\mathrm{Aut}_{R,+}(R[[1|n]]):=&\
\lim_{m\to\infty}\mathrm{Aut}_{R}(R[t|\theta^{1},\cdots,\theta^{n}]/\mathfrak{m}^{m})\\\
\mathrm{Der}_{R,+}(R[[1|n]]):=&\
\lim_{m\to\infty}\mathrm{Der}_{R}(R[t|\theta^{1},\cdots,\theta^{n}]/\mathfrak{m}^{m})\end{split}$

and a well defined _exponential_ :

$\exp:\mathrm{Der}_{R,+}(R[[1|n]])\to\mathrm{Aut}_{R,+}(R[[1|n]]).$ (2.3)

Observe that this map is surjective, since is surjective for any
$m\in\mathbb{N}$.

Denote by $\mathrm{Aut}_{0}(R[[1|n]])$ the group of automorphisms generated by
affine maps on $\\{t|\theta^{1},\dots,\theta^{N}\\}$.

It was proven [10, Lemma 6.2.1]:

######  Proposition 2.22.

The group $\mathrm{Aut}_{R}(R[[1|n]])$ is a semi-direct product of
$\mathrm{Aut}_{0}(R[[1|n]])$ and $\mathrm{Aut}_{+}(R[[1|n]])$.

An automorphism $\Phi$ is said to be generated by a vector field
$X\in\mathrm{Der}_{R,+}(R[[1|n]])$ if $\exp(X)=\Phi$ in (2.3).

####  2.4.1 The group $\mathrm{Aut}_{R}^{\delta}(R[[1|n]])$

Let $R$ be a super algebra, over $R[[1|n]]$ consider the $1|n$-free super
module $\Omega^{1}:=(\mathrm{Der}_{R}(R[[1|n]]))^{*}$ with generators
$\\{dt|d\theta^{1}\cdots d\theta^{n}\\}$, then $\mathrm{Ber}(\Omega^{1})$ has
a generator $\Delta_{0}:=[dt|d\theta^{1}\cdots d\theta^{n}]$, called _super
volume form_. There is a group homomorphism $\mathrm{Aut}_{R}(R[[1|n]])\to
End_{R}(\mathrm{Ber}(\Omega^{1}))^{*}$ given by

$\begin{split}\mathrm{Aut}_{R}(R[[1|n]])\to&\
End_{R}(\mathrm{Ber}(\Omega^{1}))^{*}\\\ \Phi\mapsto&\
\mathrm{Ber}(J\Phi)\text{, where $J\Phi$ is the Jacobian of
$\Phi$}.\end{split}$ (2.4)

This homomorphism depends on the basis chosen. Also,
$\Phi^{*}\Delta_{0}=\mathrm{Ber}(J\Phi)\Delta_{0}$.

We will denote the kernel of (2.4) by $\mathrm{Aut}_{R}^{\delta}(R[[1|n]])$.
When there is no confusion we just write $\mathrm{Aut}^{\delta}[[1|n]]$ and we
say that such automorphisms preserve the Berezinian.

For an even vector field $X\in\mathrm{Der}_{R}(R[[1|n]])$, differentiating
(2.4) we have

$L_{X}\Delta_{0}=\mathrm{sdiv}_{\Delta_{0}}(X)\Delta_{0}.$

In particular, for a vector field $X\in\mathrm{Der}_{R,+}(R[[1|n]])$ we get
the relation

$\mathrm{Ber}(\exp(X))=\exp(\mathrm{sdiv}_{\Delta_{0}}(X)),$

then, $\Phi=\exp(X)$ preserves the Berezinian if and only if
$\mathrm{sdiv}(X)=0$. This define the subalgebra

$S(1|N)_{+}:=\mathrm{Der}_{R,+}(R[[1|n]])\cap S(1|N).$

For the group $\mathrm{Aut}^{\delta}(R[[1|n]])$, we will denote by

$\mathrm{Aut}^{\delta}_{0}(R[[1|n]]):=\mathrm{Aut}_{0}(R[[1|n]])\cap\mathrm{Aut}^{\delta}(R[[1|n]])$

and

$\mathrm{Aut}^{\delta}_{+}(R[[1|n]]):=\mathrm{Aut}_{+}(R[[1|n]])\cap\mathrm{Aut}^{\delta}(R[[1|n]]),$

so we get the surjection

$S(1|N)_{+}\stackrel{{\scriptstyle\exp}}{{\longrightarrow}}\mathrm{Aut}^{\delta}_{+}(R[[1|n]]).$

From Lemma 2.22 we get:

######  Proposition 2.23.

The group $\mathrm{Aut}^{\delta}_{R}(R[[1|n]])$ is a semi-direct product of
$\mathrm{Aut}^{\delta}_{0}(R[[1|n]])$ and
$\mathrm{Aut}^{\delta}_{+}(R[[1|n]])$.

####  2.4.2 The group $\mathrm{Aut}_{R}^{\omega}(R[[1|n]])$

Another important group of automorphism is given by the automorphisms
preserving the even nondegenerate form

$\omega=dz+\theta^{1}d\theta^{1}+\cdots+\theta^{n}d\theta^{n}.$ (2.5)

Take an element $\Phi\in\mathrm{Aut}(R[[1|n]])$, we will say that
$\Phi\in\mathrm{Aut}_{R}^{\omega}(R[[1|n]])$ if $\Phi$ preserves this form up
to multiplication, in other words if $\Phi^{*}\omega=f\omega$, for some
function $f\in R[[1|n]]$. In case there is no confusion, we will write
$\mathrm{Aut}^{\omega}[[1|n]]$. Observe that $\mathrm{Aut}^{\omega}[[1|n]]$ is
a super group with the composition as multiplication.

We say that a vector field $X\in K(1|n)$ if $L_{X}\omega=f\omega$, with
$\omega$ as (2.5) for some function $f\in R[[1|n]]$ and write
$K(1|n)_{+}:=\mathrm{Der}_{R,+}(R[[1|n]])\cap K(1|n)$. We can notice that for
a vector field $X\in K(1|n)_{+}$ we have
$\exp(X)\in\mathrm{Aut}^{\omega}[[1|n]]$. The group generated by automorphisms
$\phi=\exp(X)$, $X\in K(1|n)_{+}$, is denoted by
$\mathrm{Aut}_{+}^{\omega}[[1|n]]$, also we have

$\mathrm{Aut}_{+}^{\omega}[[1|n]]=\mathrm{Aut}^{\omega}[[1|n]]\cap\mathrm{Aut}_{+}[[1|n]].$

######  Observation 2.24.

The vector fields generating automorphisms over $R[[1|n]]$ preserving (2.5),
up to multiplication, are given by:

$D^{f}=f\partial_{z}+\frac{1}{2}(-1)^{j}\sum_{i=1}^{n}(D^{i}f)D^{i},$

where $D^{i}=\theta^{i}\partial_{z}+\partial_{\theta^{i}}$, for any $f\in
R[[1|n]]_{j}$.

For $n\neq 2$, (cf. [2]), any change of coordinates that preserves $\omega$,
up to multiplication by a function, comes from fields in $K(1|n)$.

When $n=2$, there exists an exterior automorphism given by

$(z|\theta^{1},\theta^{2})\mapsto(z|\theta^{2},\theta^{1}).$

Also, for $n=2$, we have the inclusion $K(1|2)\subset S(1|2)$.

##  3 Super manifolds

######  Definition 3.1.

Let $R$ be a super algebra, we define the spectrum $\text{Spec}(R)$ as the set
of prime ideals with the Zariski topology. Over $\text{Spec}(R)$ define the
sheaf of super algebras $\mathcal{O}_{R}$ generated by

$\begin{split}\mathcal{O}_{R}(\text{Spec}R_{f})=R_{f}\end{split}$

for any non-nilpotent element $f\in R_{0}$.

A super scheme is a pair $(M,\mathcal{O}_{M})$ where $M$ is a topological
space, $\mathcal{O}_{M}$ is a sheaf of super algebras and there exists an open
covering of $M$, $\\{U_{i}\\}_{i\in I}$, such that
$(U_{i},\mathcal{O}_{M}|_{U_{i}})=(\text{Spec}R_{i},\mathcal{O}_{R_{i}})$, for
some super algebra $R_{i}$.

We define morphisms of super schemes, sheaves of modules, etc, similar to
morphisms of schemes, sheaves of modules, etc.

######  Example 3.2.

A scheme $(M,\mathcal{O}_{M})$ defines naturally a super scheme
$(M,\mathcal{O}_{M})$ where for any open set $U\subset M$, the super algebra
$\mathcal{O}_{M}(U)$ is purely even.

######  Observation 3.3.

Let $M$ a topological space and $\mathcal{F}$ a sheaf of super algebras. The
projection given in Observation 2.13 induce a morphism of sheaves given by
$\mathcal{F}(U)\to\mathcal{F}(U)_{\mathrm{rd}}$, for any open set $U\subset
M$. From this projection, for a super scheme $(M,\mathcal{O}_{M})$ we obtain a
super scheme through $\mathcal{O}_{M}\to\mathcal{O}_{M,\text{rd}}$ by the pair
$M_{\text{rd}}=(M,\mathcal{O}_{M,\mathrm{rd}})$ and a closed embedding

$M_{\text{rd}}\hookrightarrow M.$

we will say that $M_{\mathrm{rd}}$ is the _reduced super scheme_ of $M$.

######  Definition 3.4.

A smooth super curve $C$ of dimension $1|n$ is a smooth connected super
manifold $(C,\mathcal{O}_{C})$ of dimension $1|n$.

######  Observation 3.5.

Let $E$ be a locally free sheaf over a smooth scheme $M_{0}$, we obtain a
sheaf of super algebras through Observation 2.14:

$U\to S_{\mathcal{O}_{M_{0}}(U)}[E(U)],$

we will write $S_{\mathcal{O}_{M_{0}}}(E)$ and define the super scheme
$M=(M_{0},S_{\mathcal{O}_{M_{0}}}(E))$.

For a super scheme $(M,\mathcal{O}_{M})$ we will say that it _splits_ if is
isomorphic to $(M_{0},S_{\mathcal{O}_{M_{0}}}(E))$. In this case, we have an
inclusion

$\begin{split}\mathcal{O}_{M_{0}}\to\mathcal{O}_{M}=S_{\mathcal{O}_{M_{0}}}(E)\end{split}$

so, there exists a projection

$M\twoheadrightarrow M_{\text{rd}}.$

From now on, we are going to consider just super schemes that are locally
split.

For a general family of super schemes $M\to S$ if the base $S$ is not purely
even we cannot assure that there exists a projection $M\to M_{\text{rd}}$ such
that the following diagram commutes:

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M_{\text{rd}}}$$\textstyle{S}$

When such projection exists we are going to say that $M$ is _projected_.

######  Definition 3.6.

A _super manifold_ is a super scheme $(M,\mathcal{O}_{M})$, such that the
sheaf of $\mathcal{O}_{M}$-modules given by
$\text{Der}(\mathcal{O}_{M}):=\mathcal{T}_{M}$ is a locally free sheaf of
$\mathcal{O}_{M}$-modules. For a super manifold $M$, we say that it has
dimension $m|n$ if $\mathcal{T}_{M}$ has rank $m|n$. A _super curve_ is a
connected $1|n$-super manifold.

For a family $M\to S$ we say that is a _family of super manifolds_ if
$\text{Der}_{\mathcal{O}_{S}}(\mathcal{O}_{M}):=\mathcal{T}_{M/S}$ is locally
free, and that have relative dimension $m|n$ if $\mathcal{T}_{M/S}$ has rank
$m|n$. Similarly, we define a _family of curves over a super scheme_ $S$.

######  Observation 3.7.

For a closed point, $p\in M$, there exists an open set $U$ such that
$\mathcal{O}_{M}(U)=S_{\mathcal{O}_{M,\text{rd}}}(E)$, for some
$O_{M,\text{rd}}(U)$-free module $E$.

######  Example 3.8.

Let $M$ an $m|n$ a super manifold, the tangent and cotangent bundle are rank
$m|n$ locally free $\mathcal{O}_{M}$-modules. Observe, that if we have local
coordinates over an open set $U$, given by
$(z_{1},\dots,z_{m}|\theta^{1},\dots,\theta^{N})$, then the tangent space is
locally trivialized, by
$\langle\partial_{z_{1}},\dots,\partial_{z_{m}}|\partial_{\theta^{1}},\dots,\partial_{\theta^{N}}\rangle$,
and the cotangent space is locally trivialized, by $\langle
d{z_{1}},\dots,d{z_{m}}|$ $d{\theta^{1}},\dots,d{\theta^{N}}\rangle$.

######  Definition 3.9.

For a super manifold $M$ with a rank $m|n$ locally free sheaf $E$ we can
define the _Berezinean sheaf_ $\mathrm{Ber}(E)$ as follows: for an open set
$U\subset M$ such that $E(U)$ is free $\mathcal{O}_{M}(U)$-module we define
$\mathrm{Ber}(E)(U)$ as Observation 2.17.

For two open sets $U,V$ and change of coordinates $\Phi$, then the
$\mathrm{Ber}(J\Phi)$ gives us the cocycle of $\mathrm{Ber}(E)$. The sheaf
$\mathrm{Ber}(E)$ has rank $1|0$ if $n$ is even and rank $0|1$ if $n$ is odd.

Set $\mathrm{Ber}_{M}:=\mathrm{Ber}(\Omega_{M})$, and for a family $M\to S$,
consider $\mathrm{Ber}_{M/S}:=\mathrm{Ber}(\Omega_{M/S})$.

######  Example 3.10.

Let $M=(M_{0},S_{\mathcal{O}_{M_{0}}}(E))$ be a split super manifold, then we
can take the local coordinates over an open set $U\subset M_{0}$,
$\\{z_{1},\dots,z_{m}|\theta^{1},\dots,\theta^{n}\\}$, in this coordinates the
cotangent space is locally trivialized, over $U$, by $\langle
d{z_{1}},\dots,d{z_{m}}|d{\theta^{1}},\dots,d{\theta^{n}}\rangle$, so
$\mathrm{Ber}_{M}(U)=\mathcal{O}_{M}(U)[d{z_{1}},\dots,d{z_{m}}|d{\theta^{1}},\dots,d{\theta^{n}}]$.
For another coordinates $\\{w_{1},\dots,$ $w_{m}|\rho^{1},\dots,\rho^{n}\\}$
with

$\begin{split}w_{i}=&\ \phi_{i}(z_{1},\dots,z_{m})\\\ \rho^{j}=&\
\theta^{1}a_{j1}(z_{1},\dots,z_{m})+\dots+\theta^{N}a_{jN}(z_{1},\dots,z_{m}),\
j=1,\dots,N.\end{split}$

The change of coordinates of the cotangent bundle is given by

$\left[\begin{matrix}\partial_{z_{k}}\phi_{l}&B\\\
0&a_{ij}\end{matrix}\right],$

where $B=(\partial_{z_{i}}\rho^{j})$, then the change of coordinates of the
cotangent bundle is given by

$\det(\partial_{z_{k}}\phi_{l})\det(a_{ij})^{-1}.$

Using the closed embedding, $j:M_{0}\to M$, we get the isomorphism

$j^{*}\mathrm{Ber}_{M}\simeq\Omega^{m}_{M_{0}}\otimes\det E^{*}.$

Observe that in this case $\mathrm{Ber}_{M}$ is a trivial bundle when
$\Omega^{m}_{M_{0}}\simeq\det E$ as line bundles over $M_{0}$.

Let $C$ be a super curve, for a section $\Delta\in H^{0}(C,\mathrm{Ber}_{C})$
and coordinate patch $\\{(U_{i},\Phi_{i})\\}$, then there exists a family of
functions $f_{i}\in H^{0}(U_{i},\mathcal{O}_{C})$ such that

$\Delta|_{U_{i}}=f_{i}[dz_{i}|d\theta_{i}^{1}\cdots d\theta_{i}^{N}].$ (3.1)

######  Observation 3.11.

Let us consider a $1|n$-super curve $C$, a coordinate patch $U\subset C$ with
a trivialization $\Phi$, and a nonvanishing section $\Delta\in
H^{0}(U,\mathrm{Ber}_{C})$ for some $f\in H^{0}(U,\mathcal{O}_{C}^{*})$ as
(3.1). Taking an even function $F(z|\theta^{1},\cdots,\theta^{n})$, and
shrinking $U$ if is necessary, with

$\partial_{z}F(z|\theta^{1},\cdots,\theta^{n})=f(z|\theta^{1},\cdots,\theta^{n}),$

then the system of coordinates $\Psi=(w|\rho^{1},\cdots,\rho^{n})$, given by

$\begin{split}w=&\ F(z|\theta^{1},\cdots,\theta^{N}),\\\ \rho^{i}=&\
\theta^{i},\ i=1\dots,n;\end{split}$

verifies $\Delta=[dw|d\rho^{1}\cdots d\rho^{n}]$. We say that such coordinate
system $\Psi$ is _compatible_ with the section $\Delta$.

Finally, for a nonvanishing section $\Delta\in H^{0}(C,\mathrm{Ber}_{C})$,
then there exists a coordinate system of $C$, $\\{(U_{i},\Phi_{i})\\}_{i}$
such that

$\Delta|_{U_{i}}=[dz_{i}|d\theta_{i}^{1}\cdots d\theta_{i}^{N}].$

For any pair of coordinates $\Phi$, $\Psi$ defined over the same open set $U$
both compatible with $\Delta|_{U}$, then the change of coordinates preserves
the Berezinian.

For a fixed curve $C$ and a nonvanishing section $\Delta\in
H^{0}(C,\mathrm{Ber}_{C})$ we will only consider coordinates compatible with
$\Delta$.

###  3.1 $S(2)$-super curves

Fix a base super scheme $S$, we will consider curves and bundles relative to
$S$.

######  Definition 3.12.

An _$S(1|2)$ -super curve_ is a pair $(C,\Delta)$, where $C\to S$ is a super
curve and a nonvanishing section $\Delta\in H^{0}(C,\mathrm{Ber}_{C/S})$.

From Proposition 2.23, for any change of coordinates
$\Phi\in\mathrm{Aut}^{\delta}(R[[1|2]])$ there exists a divergence free field
$X\in S(1|2)_{+}$ and $T\in\mathrm{Aut}^{\delta}_{0}(R[[1|2]])$ such that
$\Phi(z|\theta^{1},\theta^{2})=\exp(X)(T(z|\theta^{1},\theta^{2}))$. Our
interest is to study such automorphisms where $X\in
S(2)_{+}:=S(2)\cap\mathrm{Der}_{R,+}(R[[1|2]])$, when this happens we write
$\Phi\in\mathrm{Aut}^{\Delta}(R[[1|2]])$ and observe that
$\mathrm{Aut}^{\Delta}(R[[1|2]])$ is a subgroup of
$\mathrm{Aut}^{\delta}(R[[1|2]])$ where

$\mathrm{Aut}^{\delta}_{0}(R[[1|2]])<\mathrm{Aut}^{\Delta}(R[[1|2]])<\mathrm{Aut}^{\delta}(R[[1|2]]).$

When there is no confusion, we write
$\mathrm{Aut}^{\Delta}[[1|2]]=\mathrm{Aut}^{\Delta}(R[[1|2]])$.

######  Definition 3.13.

An _$S(2)$ -super curve_ is an $S(1|2)$-super curve $(C,\Delta)$ such that
there exists a system of coordinates $\\{U_{i},\Phi_{i}\\}$ compatible with
$\Delta$ and change of coordinates
$\Phi_{ij}=\Phi_{i}\circ\Phi^{-1}_{j}\in\mathrm{Aut}^{\Delta}[[1|2]]$. We say
that a family of curves $C\to S$ is a family of $S(2)$-super curves, if for
any $y\in S$, $C_{y}$ is an $S(2)$-super curve.

Over a curve $C\to S$ with a nonvanishing section $\Delta\in
H^{0}(C,\mathrm{Ber}_{C/S})$, over an open set we can define the space of
vector fields:

$S(1|2)(U):=\\{X\in\mathcal{T}_{C}(U):\mathrm{sdiv}_{\Delta}X=0\\},$

and

$S(2)(U):=[S(1|2)(U),S(1|2)(U)].$

Observe that such spaces do not define a sheaf of $\mathcal{O}_{C}$-modules.
They define, however, a sheaf of $\pi^{*}\mathcal{O}_{S}$-modules.

Finally, we get a $\pi^{*}\mathcal{O}_{S}$-module $K$ defined over an open set
$U\subset C_{0}$:

$K(U):=\frac{S(1|2)(U)}{S(2)(U)}.$

On the other hand, we have the isomorphisms

$\begin{split}\exp:S(1|2)_{+}\to\mathrm{Aut}^{\delta}_{+}(R[[1|n]]),\\\
\exp:S(2)_{+}\to\mathrm{Aut}^{\Delta}_{+}(R[[1|n]]),\end{split}$

and $\mathrm{Aut}^{\delta}_{0}(R[[1|n]])=\mathrm{Aut}^{\Delta}_{0}(R[[1|n]])$,
so, we have the isomorphism:

$\exp:\frac{S(1|2)}{S(2)}\to\frac{\mathrm{Aut}^{\delta}(R[[1|n]])}{\mathrm{Aut}^{\Delta}(R[[1|n]])}\simeq\mathbb{G}_{a}.$
(3.2)

######  Observation 3.14.

For an $S(1|2)$-super curve $(C,\Delta)$ we can construct the bundle of
coordinates preserving the Berezinian, $\mathrm{Aut}^{\delta}_{C}$, considered
as the set of pairs $(Z,\Phi)$ for $Z$ a $S$-point in $C$ and $\Phi$ a local
system of coordinates compatible with the section $\Delta$. This bundle is an
$\mathrm{Aut}^{\delta}(R[[1|2]])$-bundle, and observe that the quotient:

$\mathrm{Aut}^{\Delta}[[1|2]]\backslash\mathrm{Aut}^{\delta}_{C}\to C$ (3.3)

is an $\mathbb{G}_{a}$-bundle. From (3.2), we get that
$\mathrm{Aut}^{\Delta}[[1|2]]\backslash\mathrm{Aut}^{\delta}_{C}$ is
isomorphic to $K$.

Now, we can reformulate the definition of $S(2)$-super curves:

######  Proposition 3.15.

An $S(2)$-super curve is an $S(1|2)$-super curve $(C,\Delta)$ such that the
$\mathbb{G}_{a}$-bundle (3.3) is trivial.

###### Proof.

Observe that the bundle (3.3) is trivial if and only if it has a section. In
this case, a section is a covering with trivializations
$\\{(U_{i},\Phi_{i})\\}_{i}$ such that the change of coordinates
$\Phi_{ij}:=\Phi_{j}\circ\Phi_{i}^{-1}\in\mathrm{Aut}^{\Delta}(R[[1|2]])$.

Finally, the bundle is trivial if and only if there exists a covering for $C$
with trivializations $\\{(U_{i},\Phi_{i})\\}_{i}$ compatible with $\Delta$
such that the change of coordinates
$\Phi_{ij}\in\mathrm{Aut}^{\Delta}[[1|2]]$, that is, $(C,\Delta)$ is an
$S(2)$-super curve. ∎

######  Observation 3.16.

For a $1|2$-super curve over an even base $S$, $C\to S$, it is shown in [11]
that there exists a similar class that measures if the curve is split. Suppose
that $C$ is a $1|2$-super curve over a point with reduced curve $C_{0}$, then
we have a sequence

$0\to\mathcal{J}\to\mathcal{O}_{C}\to\mathcal{O}_{C_{0}}\to 0,$

for which we have the inclusion $C_{0}\stackrel{{\scriptstyle j}}{{\to}}C$.
Now considering the curve $\overline{C}=(C_{0},\mathcal{O}_{C,0})$ and the
inclusion $C_{0}\stackrel{{\scriptstyle j}}{{\to}}\overline{C}$ we get the
sequence of sheaves of algebras over $\overline{C}$:

$0\to\det\mathcal{F}\to\mathcal{O}_{C,0}\to\mathcal{O}_{C_{0}}\to 0,$ (3.4)

for the rank 2 bundle $\mathcal{F}=\mathcal{J}/\mathcal{J}^{2}$, in
$\det\mathcal{F}$ we consider the zero multiplication. Taking local splits
$\pi_{i}:\mathcal{O}_{C_{0}}(U_{i})\to\mathcal{O}_{C,0}(U_{i})$ in (3.4) we
can define

$\omega_{ij}=\pi_{i}|_{U_{i}\cap U_{j}}-\pi_{j}|_{U_{i}\cap U_{j}}$ (3.5)

observe that

$\begin{split}\omega_{ij}(fg)=&\pi_{i}(fg)-\pi_{j}(fg)\\\
=&\pi_{i}(f)\pi_{i}(g)-\pi_{j}(f)\pi_{i}(g)\\\
=&\pi_{i}(f)\omega_{ij}(g)+\omega_{ij}(f)\pi_{i}(g)\end{split}$

so $\omega_{ij}\in\mathcal{T}_{C_{0}}\otimes\det\mathcal{F}(U_{ij})$.
Additionally, $\omega_{ij}$ verifies the cocycle condition:

$\omega_{ij}+\omega_{jk}+\omega_{ki}=(\pi_{i}|_{U_{i}\cap U_{j}\cap
U_{k}}-\pi_{j}|_{U_{i}\cap U_{j}\cap U_{k}})+(\pi_{j}|_{U_{i}\cap U_{j}\cap
U_{k}}-\pi_{k}|_{U_{i}\cap U_{j}\cap U_{k}})\\\ +(\pi_{k}|_{U_{i}\cap
U_{j}\cap U_{k}}-\pi_{i}|_{U_{i}\cap U_{j}\cap U_{k}})=0$

so $\\{\omega_{ij}\\}\in
H^{1}(C_{0},T_{C_{0}}\otimes\det\mathcal{F}(U_{ij}))$.

For an $S(1|2)$-super curve we have $\det\mathcal{F}=\Omega_{C_{0}}$, then
$\\{\omega_{ij}\\}\in
H^{1}(C_{0},T_{C_{0}}\otimes\Omega_{C_{0}})=H^{1}(C_{0},\mathcal{O}_{C_{0}})$.
Let $\\{\Phi_{i}=(z_{i}|\theta^{1}_{i},\theta^{2}_{i})\\}$ local coordinates
over $C$. For the change of coordinates $\phi_{ij}$ we get

$\begin{split}z_{j}=&\
F_{ij}(z_{i})+G_{ij}(z_{i})\theta^{1}_{i}\theta^{2}_{i}\\\ \theta^{1}_{j}=&\
\theta^{1}_{i}a_{11}(z_{i})+\theta^{2}_{i}a_{12}(z_{i})\\\ \theta^{2}_{j}=&\
\theta^{1}_{i}a_{21}(z_{i})+\theta^{2}_{i}a_{22}(z_{i}).\end{split}$

Since $C$ has a trivial Berezinian, then
$G_{ij}=\lambda_{ij}\partial_{z_{i}}F_{ij}$, for $\lambda_{ij}$ a constant,
then we have

$\begin{split}z_{j}=&\
F_{ij}(z_{i})+\lambda_{ij}\partial_{z_{i}}F_{ij}(z_{i})\theta^{1}_{i}\theta^{2}_{i}=F_{ij}(z_{i}+\lambda_{ij}\theta^{1}_{i}\theta^{2}_{i})\\\
\theta^{1}_{j}=&\
\theta^{1}_{i}a_{11}(z_{i})+\theta^{2}_{i}a_{12}(z_{i})=g^{1}_{ij}(z_{i}|\theta^{1}_{i},\theta^{2}_{i})\\\
\theta^{2}_{j}=&\
\theta^{1}_{i}a_{21}(z_{i})+\theta^{2}_{i}a_{22}(z_{i})=g^{2}_{ij}(z_{i}|\theta^{1}_{i},\theta^{2}_{i}).\end{split}$

From the generators given in Observation 2.20, we get that
$\widetilde{\Phi}_{ij}=(F_{ij}|g^{1}_{ij},g^{2}_{ij})\in\mathrm{Aut}^{\Delta}[[1|2]]$
and
$z_{i}+\lambda_{ij}\theta^{1}\theta^{2}_{i}=\exp(\lambda_{ij}\theta_{i}^{1}\theta^{2}_{i}\partial_{z_{i}})(z_{i})$,
then
$\Phi_{ij}=\widetilde{\Phi}_{ij}\circ\exp(\lambda_{ij}\theta_{i}^{1}\theta^{2}_{i}\partial_{z_{i}})$,
that is $\exp(\lambda_{ij}\theta_{i}^{1}\theta^{2}_{i}\partial_{z_{i}})$ gives
the cocycle in (3.3).

The relation of (3.5) with the class (3.3) is the following:

$\omega_{ij}(f)=\lambda_{ij}\theta_{i}^{1}\theta_{i}^{2}\partial_{z_{i}}f$

for local coordinates $(z_{i}|\theta_{i}^{1},\theta_{i}^{2})$ over $U_{i}$.

In [11] it is proved that $C\to S$ is projected if and only if
$\\{\omega_{ij}\\}\in H^{1}(C,\mathcal{O}_{C/S})$ vanishes. Then we obtain

######  Theorem 3.17.

Every $S(2)$-super curve over a purely even base $S$ is split.

In order to get a geometric interpretation of this,consider a $1|2$-super
curve $C$, the inclusion $C_{0}\stackrel{{\scriptstyle j}}{{\to}}\overline{C}$
and the space of differentials over $\Omega_{\overline{C}}$, we obtain that
$j^{*}\Omega_{\overline{C}}$ is a rank 2 bundle over $C_{0}$ with a projection
$j^{*}\Omega_{\overline{C}}\to\Omega_{C_{0}}\to 0$. Actually, we get the
sequence of $\mathcal{O}_{C_{0}}$-modules:

$0\to\det\mathcal{F}\to j^{*}\Omega_{\overline{C}}\to\Omega_{C_{0}}\to 0.$
(3.6)

As an extension of $\mathcal{O}_{C_{0}}$-modules, (3.6), is defined by an
element of

$\text{Ext}^{1}(\Omega_{C_{0}},\det\mathcal{F})=H^{1}(\Omega_{C_{0}}^{*}\otimes\det\mathcal{F})=H^{1}(\mathcal{T}_{C_{0}}\otimes\det\mathcal{F}),$

and such element is the class $\\{\omega_{ij}\\}$ defined above.

Now, if $\det\mathcal{F}=\Omega_{C_{0}}$, then we have the sequence

$0\to\Omega_{C_{0}}\to j^{*}\Omega_{\overline{C}}\to\Omega_{C_{0}}\to 0.$
(3.7)

This occurs when $C$ is a $S(1|2)$ curve, but to distinguish the element $K\in
H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})\subseteq
H^{1}(C_{0},\mathcal{O}_{C_{0}})$, where $\pi:C\to S$, we have to notice that
the sequence (3.7) fits in the following diagram:

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(3.8)

where $L\otimes\Omega_{C_{0}}=j^{*}\Omega_{\overline{C}}$ and $L$ as an
extension of $\pi^{*}\mathcal{O}_{S}$-modules represents the class
$\Gamma_{C}\in H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$.

Finally, in [12] and [7] it is proved that each $1|2$ super curve, over a
point, is defined by the data of $(C_{0},\mathcal{F},\\{\omega_{ij}\\})$,
where $\\{\omega_{ij}\\}$ represents an extension of
$\mathcal{O}_{C_{0}}$-modules:

$0\to\det\mathcal{F}\to\mathcal{L}\to\Omega_{C_{0}}\to 0.$

For an $S(1|2)$ we need $(C_{0},\mathcal{F},\Gamma_{C})$, where $\Gamma_{c}$
represents an extension of $\pi^{*}\mathcal{O}_{S}$-modules

$0\to\pi^{*}\mathcal{O}_{S}\to L\to\pi^{*}\mathcal{O}_{S}\to 0$

which gives rise to the diagram (3.8).

###  3.2 $SUSY$-super curves

######  Definition 3.18.

A $1|n$-super curve $C$ with a covering and coordinate systems
$\\{\phi_{i}\\}_{i}$ such that any change of coordinates verifies
$\phi_{ij}=\Phi_{i}\circ\Phi^{-1}_{j}\in\mathrm{Aut}^{\omega}[[1|n]]$ we will
say that $C$ is a _$SUSY_{n}$ -super curve_.

Observe that the local form (2.5) in coordinates
$\phi_{i}=(z_{i}|\theta^{1}_{i},\dots,\theta^{n}_{i})$

$\omega_{i}=dz_{i}+\theta^{1}_{i}d\theta^{1}_{i}+\cdots+\theta^{n}_{i}d\theta^{n}_{i}$

is well defined, up to multiplication by a function over $C$.

Also, we can define an $SUSY_{n}$-super structure over the $1|n$-super curve
$C$ as a locally free subsheaf $E\subset\mathcal{T}_{C}$ of rank $0|n$, for
which the Frobenius form

$E\otimes E\to\mathcal{T}_{C}/E$

is nondegenerate and split, i.e., it locally has an isotropic direct subsheaf
of maximal possible rank $k$ for $n=2k$ or $2k+1$ (cf. 2).

######  Observation 3.19.

For a $1|2$-super curve $C$, if each change of coordinates comes from a
$K(1|2)$ field we say that such curve is orientable. Oriented curves are also
characterized by the fact that the distribution $E$ decomposes as

$E=\mathscr{L}\oplus\mathscr{L}^{*}\otimes\Omega_{C},$

for some line bundle $\mathscr{L}$ over $C_{0}$.

Observe that in the non-oriented case such bundle $\mathscr{L}$ only exists
locally.

Since $K(1|2)\subseteq S(2)$, then any change of coordinates that comes from a
field on $K(1|2)$ preserves the Berezinian and actually any orientable
$SUSY_{2}$-super curve is an $S(2)$-super curve.

######  Observation 3.20.

Suppose that $N=2n$, over $R[[1|2n]]$ consider the change of coordinates

$\begin{split}w=&z+i\left(\theta^{1}\theta^{2}+\cdots+\theta^{2n-1}\theta^{2n}\right),\\\
\rho^{j}=&-i(-\theta^{2j-1}+i\theta^{2j}),\ j=1\dots,n,\\\
\eta^{j}=&-i(\theta^{2j-1}+i\theta^{2j}),\ j=1\dots,n,\end{split}$

we obtain that $\omega=dz+\theta^{1}d\theta^{1}+\cdots+\theta^{n}d\theta^{n}$
changes as

$\widetilde{\omega}=dz+\rho^{1}d\eta^{1}+\cdots+\rho^{n}d\eta^{n}$ (3.9)

so, the group of automorphisms of $R[[1|2n]]$ that preserve $\omega$ up to
multiplication by a function coincides with the group of automorphisms of

$R[[w|\rho^{1},\cdots,\rho^{n},\eta^{1},\cdots,\eta^{n}]]$

that preserve (3.9) up to multiplication by a function.

Observe that for a $1|2n$-super curve $C$ for which (3.9) is well defined, up
to multiplication by a function, then $C$ is a $SUSY$-super curve, the
converse is not necessarily true.

Consider the inclusion $R[[1|n]]\hookrightarrow R[[1|2n]]$ given by the
identification

$R[[1|2n]]=R[[1|n]][\rho^{1},\dots,\rho^{n}].$

Let $\Phi=(F|\phi^{1},\dots,\phi^{n})\in\mathrm{Aut}[[1|n]]$ and consider the
super function
$\widetilde{\Phi}=(F|\phi^{1},\dots,\phi^{n},\eta^{1},\dots,\eta^{n})$. The
pullback of the form (3.9) is given by

$\begin{split}\widetilde{\Phi}^{*}(dz+\rho^{1}d\theta^{1}+\dots+\rho^{n}d\theta^{n})=&\partial_{z}Fdz-\partial_{\theta^{1}}Fd\theta^{1}-\dots-\partial_{\theta^{n}}Fd\theta^{n}\\\
&+\eta^{1}(\partial_{z}\phi^{1}dz+\partial_{\theta^{1}}\phi^{1}d\theta^{1}+\dots+\partial_{\theta^{n}}\phi^{1}d\theta^{n})+\cdots\\\
&\hskip
25.6073pt+\eta^{n}(\partial_{z}\phi^{n}dz+\partial_{\theta^{1}}\phi^{n}d\theta^{1}+\dots+\partial_{\theta^{n}}\phi^{n}d\theta^{n})\\\
=&(\partial_{z}F+\eta^{1}\partial_{z}\phi^{1}+\cdots+\eta^{n}\partial_{z}\phi^{n})dz\\\
&\hskip
11.38092pt+(-\partial_{\theta^{1}}F+\eta^{1}\partial_{z}\phi^{1}+\cdots+\eta^{n}\partial_{z}\phi^{n})d\theta^{1}+\cdots\\\
&\hskip
39.83368pt+(-\partial_{\theta^{n}}F+\eta^{1}\partial_{z}\phi^{1}+\cdots+\eta^{n}\partial_{z}\phi^{n})d\theta^{n}.\end{split}$

Then, defining the differential operators
$D_{j}=\rho^{j}\partial_{z}+\partial_{\theta^{j}}$, the function
$\widetilde{\Phi}\in\mathrm{Aut}^{\omega}[[1|2n]]$ if and only if

$D_{i}F=\eta^{1}D_{i}\phi^{1}+\cdots+\eta^{n}D_{i}\phi^{n},\ i=1,\dots,n.$
(3.10)

Since the matrix $(D_{i}\phi^{j})_{ij}$ is invertible, we get:

$\left[\begin{matrix}D_{1}\phi^{1}&\cdots&D_{1}\phi^{n}\\\ \vdots&&\vdots\\\
D_{n}\phi^{1}&\cdots&D_{n}\phi^{n}\end{matrix}\right]^{-1}\left[\begin{matrix}D_{1}F\\\
\vdots\\\ D_{n}F\end{matrix}\right]=\left[\begin{matrix}\eta^{1}\\\ \vdots\\\
\eta^{n}\end{matrix}\right].$ (3.11)

With these coordinates,
$\widetilde{\Phi}=(F|\phi^{1},\dots,\phi^{n},\eta^{1},\dots,\eta^{n})$ makes
the following diagram commutes

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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42.22511pt\raise-33.38889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\Phi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.0029pt\raise-40.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.0029pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R[[1|2n]]}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(3.12)

and we obtain the inclusion of groups:

$\mathrm{Aut}[[1|n]]\stackrel{{\scriptstyle
j}}{{\hookrightarrow}}\mathrm{Aut}^{\omega}[[1|2n]].$ (3.13)

Also, we obtain an inclusion of Lie algebras given by

$\mathrm{Der}_{R}(R[[1|n]])\stackrel{{\scriptstyle\widehat{j}}}{{\hookrightarrow}}K(1|2n),$

in such way that for $X\in\mathrm{Der}_{R,+}(R[[1|n]])$ we obtain
$j(\exp(X))=\exp(\widehat{j}(X))$.

######  Proposition 3.21.

Any $1|n$-super curve has an $SUSY_{2n}$-super curve associated.

###### Proof.

Given a super curve $C$, consider the $\mathrm{Aut}[[1|n]]$-principal bundle
$\mathrm{Aut}_{C}$, then we can construct an
$\mathrm{Aut}^{\omega}[[1|2n]]$-principal bundle given by

$\mathrm{Aut}^{\omega}[[1|2n]]\times_{\mathrm{Aut}[[1|n]]}\mathrm{Aut}_{C}$

for the inclusion $\mathrm{Aut}[[1|n]]\subset\mathrm{Aut}^{\omega}[[1|2n]]$
given in (3.13).

The structure of $1|n$-super curve give us family of local sections of the
bundle $\mathrm{Aut}_{C}\to C$, $\\{\phi_{i}\\}_{i\in I}$; then the family of
local sections $\\{\Phi_{i}=(1,\phi_{i})\\}_{i\in I}$ gives us a family of
$K(1|2n)$-super curves. ∎

The curve obtained in Proposition 3.21 is going to be denoted by
$\widetilde{C}$. If we have an atlas
$\\{\phi_{i}=(z|\theta^{1},\theta^{2})\\}_{i\in I}$ over $C$ with cocycles
$\phi_{ij}=\Phi_{i}\circ\Phi^{-1}_{j}$, we construct the atlas over
$\widetilde{C}$ given by
$\\{\widetilde{\phi}_{i}:=(z|\theta^{1},\theta^{2},\rho^{1},\rho^{2})\\}$,
with cocycles $\widetilde{\phi}_{i}\widetilde{\phi}^{-1}_{j}=j(\phi_{ij})$,
with $j$ given in (3.13).

######  Observation 3.22.

In the previous construction, we get a projection locally given by

$(z|\theta^{1},\dots,\theta^{n},\rho^{1},\dots,\rho^{n})\mapsto(z|\theta^{1},\dots,\theta^{n}).$

By construction, the projection
$\widetilde{C}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}C$ is well
defined. For any point $p\in C$ the fiber has dimension $0|n$.

######  Observation 3.23.

There exists a geometric description of this fact given by [8]. The space
$\widetilde{C}$ is described by the space of $0|n$-subspaces of $T_{p}C$ for
any $p\in C$. The $SUSY$-super structure for a point $(p,E)$ is given by the
distribution $\widetilde{E}\subset\mathcal{T}_{(p,E)}\widetilde{C}$ defined by
the local form $dz+\rho^{1}d\theta^{1}+\dots+\rho^{n}d\theta^{n}$ for
$p=(z|\theta^{1},\dots,\theta^{n})$. Locally, $\widetilde{E}$ is generated by
$\\{\partial_{\rho^{1}},\dots,\partial_{\rho^{n}},\rho^{1}\partial_{z}+\partial_{\theta^{1}},\dots,\rho^{n}\partial_{z}+\partial_{\theta^{n}}\\}$.

In this context, the projection
$\widetilde{C}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}C$ is given by
$(p,E)\mapsto p$ and the distribution $\widetilde{E}$ is given by
$d\pi^{-1}(E)$, for $T_{(p,E)}\widetilde{C}\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}T_{p}C$.

For $SUSY_{n}$-super curve $C\to S$ the operators
$D^{i}=\theta^{i}\partial_{z}+\partial_{i}$ define a $0|n$-distribution over
$T_{C/S}$ with
$D_{\alpha}^{i}=(D_{\alpha}^{i}\phi_{\alpha,\beta}^{1})D_{\beta}^{i}+\cdots+(D_{\alpha}^{i}\phi_{\alpha,\beta}^{n})D_{\beta}^{i},\
i=1,\dots,n$. For the change of coordinates $(F|\phi^{1},\dots,\phi^{n})$ the
operators $D^{i}$, for $i=1,\dots,n$, verify

$D^{i}F=\phi^{1}D^{i}\phi^{1}+\cdots+\phi^{n}D^{i}\phi^{n},\ i=1,\dots,n.$
(3.14)

######  Observation 3.24.

The $SUSY_{2n}$-super curve $\widetilde{C}$ associated to the $1|n$-super
curve $C$ has a rank $0|n$ bundle locally generated by the local fields
$D^{i}=\partial_{\rho^{i}}$, $i=1,\dots,n$. Reciprocally, suppose that $C$ is
a $SUSY_{2n}$-super curve with the local coordinates
$(w|\theta^{1},\cdots,\theta^{2n})$ with a change of coordinates
$(F|\phi^{1},\cdots,\phi^{n})$, introducing the new variables

$\begin{split}w=&\
z+i\left(\theta^{1}\theta^{2}+\cdots+\theta^{2n-1}\theta^{2n}\right)\\\
\zeta^{j}=&-i(-\theta^{2j-1}+i\theta^{2j}),\ j=1,\dots,n.\\\
\rho^{j}=&-i(\theta^{2j-1}+i\theta^{2j}),\ j=1,\dots,n.\end{split}$

The change of coordinates

$\begin{split}G=&F+i\left(\phi^{1}\phi^{2}+\cdots+\phi^{2n-1}\phi^{2n}\right)\\\
\psi^{j}=&-i(-\phi^{2j-1}+i\phi^{2j}),\ j=1,\dots,n,\\\
\eta^{j}=&-i(\phi^{2j-1}+i\phi^{2j}),\ j=1,\dots,n,\end{split}$

and considering the operators $D^{j}_{\pm}=\frac{1}{2}\left\\{D^{2j}\pm
iD^{2j-1}\right\\}$, the equation (3.14) reads:

$D^{j}_{\pm}G=\eta^{1}D^{j}_{\pm}\psi^{1}+\cdots+\eta^{n}D^{j}_{\pm}\psi^{n}.$
(3.15)

This induces the rank $0|2n$-distribution

$D^{j}_{\alpha,\pm}=\sum_{k=1}^{n}\left((D^{j}_{\alpha,\pm}\psi^{k})D^{j}_{\beta,-}+(D^{j}_{\alpha,\pm}\eta^{k})D^{j}_{\beta,+}\right)$
(3.16)

and if the following equations hold

$D^{j}_{+}\psi^{k}=0,\qquad j,k=1,\dots,n;$ (3.17)

then we can define the $1|n$-super curve by the coordinates
$(w|\zeta^{1},\dots,\zeta^{n})$. The conditions described in equation (3.17)
are equivalent to having the $0|n$-distribution locally defined by
$D^{j}_{+}$, $j=1,\dots,n$.

Let $C$ be a $SUSY_{4}$-super curve. Taking the
$\mathrm{Aut}^{\omega}[[1|4]]$-principal bundle $\mathrm{Aut}_{C}\to C$, we
obtain that it comes from a $1|2$-super curve if and only if the bundle
$\mathrm{Aut}[[1|2]]\backslash\mathrm{Aut}_{C}$ has a global section.

######  Example 3.25.

The $1|1$-super curves induce what are called oriented $SUSY_{2}$-super
curves. For a general $SUSY_{2}$-super curve and change of coordinates
$(F|\psi,\eta)$ equations (3.16) are:

$\begin{split}D_{\alpha,+}=&D_{\alpha,+}\psi D_{\beta,-}+D_{\alpha,+}\eta
D_{\beta,+},\\\ D_{\alpha,-}=&D_{\alpha,-}\psi D_{\beta,-}+D_{\alpha,-}\eta
D_{\beta,+},\end{split}$

and since the matrix

$\left[\begin{matrix}D_{\alpha,+}\psi&D_{\alpha,+}\eta\\\
D_{\alpha,-}\psi&D_{\alpha,-}\eta\end{matrix}\right]$

is invertible, then $D_{\alpha,-}\psi D_{\alpha,+}\eta\neq 0$. If (3.17)
holds, differentiating equation (3.15), we get

$\begin{split}D_{\alpha,-}D_{\alpha,-}F=&\ D_{\alpha,-}\left(\eta
D_{\alpha,-}\psi\right),\\\ =&\ D_{\alpha,-}\eta D_{\alpha,-}\psi-\eta
D_{\alpha,-}D_{\alpha,-}\psi;\end{split}$

since $D_{\alpha,-}D_{\alpha,-}=0$, we obtain that $D_{\alpha,-}\eta
D_{\alpha,-}\psi=0$, then $D_{\alpha,-}\eta$, so there exists another bundle
defined by $D_{-}$ that for a change of coordinates $(F|\psi,\eta)$ we get

$D_{\alpha,-}=D_{\alpha,-}\psi D_{\beta,-}.$

This line bundle induces another curve $\widehat{C}$ defined by the
coordinates:

$\begin{split}\widehat{z}=&z-\theta\rho,\\\ \widehat{\rho}=&\rho,\end{split}$

this curve is called the “dual” curve associated to $C$. The situation is
described in [2].

######  Example 3.26.

Over the affine plane $\mathbb{C}^{1|4}$ consider the following relations

  1. 1.

$A(z|\theta^{1},\theta^{2},\theta^{3},\theta^{4})=(z+1|\theta^{1},\theta^{2},\theta^{3},\theta^{4})$,

  2. 2.

$B(z|\theta^{1},\theta^{2},\theta^{3},\theta^{4})=(z+\tau-2\theta^{1}\theta^{2}\theta^{3}\theta^{4}|\theta^{1}+\theta^{2}\theta^{3}\theta^{4},\theta^{2}-\theta^{1}\theta^{3}\theta^{4},\theta^{3}+\theta^{1}\theta^{2}\theta^{4},\theta^{4}-\theta^{1}\theta^{2}\theta^{3})$.

Where $\tau$ is a non null even parameter. The quotient
$\mathbb{C}^{1|4}/\langle A,B\rangle$ is an elliptic curve
$\widetilde{\mathbb{T}}_{\tau}$ with a $SUSY_{4}$-super structure. Its tangent
bundle $T_{\widetilde{\mathbb{T}}_{\tau}}$ does not have a subbundle of
dimension $0|2$ then this curve does not come from a $1|2$-super curve.

######  Observation 3.27.

For $n=2$ the change of coordinates for our new variables
$(\rho^{1},\rho^{2})$ are given by the equation:

$\left[\begin{matrix}\eta^{1}\\\
\eta^{2}\end{matrix}\right]=A^{-1}\left[\begin{matrix}\partial_{1}F\\\
\partial_{2}F\end{matrix}\right]+\mathrm{Ber}(\Phi)\left(\left[\begin{matrix}\partial_{2}\phi^{2}&-\partial_{1}\phi^{2}\\\
-\partial_{2}\phi^{1}&\partial_{1}\phi^{1}\end{matrix}\right]\left[\begin{matrix}\rho^{1}\\\
\rho^{2}\end{matrix}\right]+\left[\begin{matrix}\partial_{z}\phi^{2}\\\
-\partial_{z}\phi^{1}\end{matrix}\right]\rho^{1}\rho^{2}\right),$ (3.18)

where $A=(\partial_{i}\phi^{j})_{i,j}$ and this gives a structure of
$SUSY_{4}$-super curve.

More specifically, observe that the local coordinates
$\rho^{1}\rho^{2},\rho^{1},\rho^{2}$ define a $1|2$ bundle over $C$ and since

$\left[\begin{matrix}\eta^{1}\eta^{2}\\\ \eta^{2}\\\
-\eta^{1}\end{matrix}\right]=\det
A^{-1}\left[\begin{matrix}\partial_{1}F\partial_{2}F\\\
\left[\begin{matrix}\partial_{2}\phi^{1}&\partial_{1}\phi^{1}\\\
\partial_{2}\phi^{2}&\partial_{1}\phi^{2}\end{matrix}\right]\left[\begin{matrix}-\partial_{1}F\\\
\partial_{2}F\end{matrix}\right]\end{matrix}\right]+\mathrm{Ber}(\Phi)J\Phi\left[\begin{matrix}\rho^{1}\rho^{2}\\\
\rho^{2}\\\ -\rho^{1}\end{matrix}\right].$

Then the sheaf $\mathcal{A}_{\widetilde{C}}$ that defines $\widetilde{C}$ is
given by the extension of $\mathcal{O}_{C}$ by
$\mathrm{Ber}_{C}\otimes\Omega_{C}$:

$0\to\mathcal{O}_{C}\to\mathcal{A}_{\widetilde{C}}\to\mathrm{Ber}_{C}\otimes\Omega_{C}\to
0.$

When our curve is split, and associated to the bundle $E$, then
$\widetilde{C}$ is a split curve associated the reduced curve $C_{0}$ and

$W=E\oplus(E^{*}\otimes\Omega_{C_{0}}),$

where $\Omega_{C_{0}}$ is the canonical bundle over $C_{0}$. In this special
case, we can consider the _dual curve_ as the split curve associated to
$E^{*}\otimes\Omega_{C_{0}}$.

##  4 $S(2)$-super curves and $SUSY_{4}$-super curves

In this section we consider the adjoint action
$\mathrm{Ad}_{\alpha}:\mathrm{Aut}^{\omega}[[1|4]]\to\mathrm{Aut}^{\omega}[[1|4]]$
given by $\alpha\in\mathrm{Aut}^{\omega}[[1|4]]$:

$\alpha(z|\theta^{1},\theta^{2},\rho^{1},\rho^{2}):=(z-\theta^{1}\rho^{1}-\theta^{2}\rho^{2},\rho^{1},\rho^{2},\theta^{1},\theta^{2}).$

Observe that $\alpha$ is an involution and $\mathrm{Ad}_{\alpha}$ fixes the
subspace $\mathrm{Aut}^{\omega}[[1|2]]\subset\mathrm{Aut}^{\omega}[[1|4]]$.

For the automorphism $\mathrm{Ad}_{\alpha}$ we have that
$\mathrm{Aut}[[1|2]]\cap\mathrm{Ad}_{\alpha}\left(\mathrm{Aut}^{\delta}[[1|2]]\right)=\mathrm{Aut}^{\Delta}[[1|2]]$.
Then we have the commutative diagram

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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.5627pt\raise-40.89275pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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(4.1)

Let $C$ be an $S(1|2)$-super curve, over the
$\mathrm{Aut}^{\delta}[[1|2]]$-principal bundle, $\mathrm{Aut}^{\delta}_{C}$,
construct the $\mathrm{Aut}^{\omega}[[1|4]]$-principal bundle,
$\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C}$,
after this we obtain a morphism given by

$\begin{split}\mu:\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C}\to&\
\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C}\\\
(g,(x,\Phi))\mapsto&\ (\alpha\circ g,(x,\Phi)).\end{split}$ (4.2)

If $C$ is an $S(2)$-super curve, then the bundle
$K=\mathrm{Aut}^{\delta}_{C}/\mathrm{Aut}^{\Delta}\to C$ is trivial, we have a
global section $s:C\to K$, defined by local sections $\\{\phi_{i}\\}_{i}$ of
the bundle $\mathrm{Aut}^{\delta}_{C}\to C$ such that
$\phi_{i}\circ\phi_{j}^{-1}\in\mathrm{Aut}^{\Delta}[[1|2]]$. Using (4.2), we
get local sections of
$\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C}\to
C$ by
$\\{\mathrm{Ad}_{\alpha}\left((1,\phi_{i})\right)=(\alpha,\phi_{i})\\}_{i}$,
since $\phi_{i}\circ\phi_{j}^{-1}\in\mathrm{Aut}^{\Delta}[[1|2]]$, then
$\mu(\phi_{i}\circ\phi_{j}^{-1})\in\mathrm{Aut}^{\Delta}[[1|2]]$. In
particular, such sections give us a global section of
$\mathrm{Aut}^{\Delta}\backslash\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C}\to
C$. Then, we obtain another family of $S(2)$-super curves $\widehat{C}\to S$.
Such family is called dual family of curves, or simply dual curve.

The bundle $\mathrm{Ad}_{\alpha}(K)$ defines a family of $S(2)$-super curves
if $C\to S$ is a family of $S(2)$-super curves.

######  Observation 4.1.

For a $\Phi\in\mathrm{Aut}^{\omega}[[1|2]]$ we have $\mu(\Phi)=\Phi$, then for
a $SUSY_{2}$-super curve in the diagram $\eqref{nomaestro}$ we have the dual
curve is isomorphic to the original one.

Similar to (3.15) we obtain:

######  Proposition 4.2.

Given a family of $S(1|2)$-super curves $C\to S$, the image
$\mathrm{Ad}_{\alpha}(K)$ in (4.2) defines a family of $1|2$-super curves if
and only if $C\to S$ is a family of $S(2)$-super curves. In such case, we
obtain the family of dual curves.

###### Proof.

The bundle $\mathrm{Ad}_{\alpha}(K)$ defines a $1|2$-super curve if and only
if the projection of $\mathrm{Ad}_{\alpha}(K)$ over
$\mathrm{Aut}[[1|2]]\backslash(\mathrm{Aut}^{\omega}[[1|4]]\times_{\mathrm{Aut}^{\delta}[[1|2]]}\mathrm{Aut}^{\delta}_{C})$
has a global section, since
$\mathrm{Aut}[[1|2]]\cap\mu\left(\mathrm{Aut}^{\delta}[[1|2]]\right)=\mathrm{Aut}^{\Delta}[[1|2]]$
then such projection is isomorphic to $K$, then $\mathrm{Ad}_{\alpha}(K)$ is
trivial if and only if $K$ is trivial, that is if $C\to S$ is an $S(2)$-super
curve. ∎

Then we obtain:

######  Theorem 4.3.

There exists an involution $\mu$ of the moduli space $\mathcal{M}_{S(2)}$ of
$S(2)$-super curves. The fixed point set of $\mu$ consists of the moduli space
$\mathcal{M}_{K(1|2)}$ of orientable $SUSY_{2}$-super curves.

######  Observation 4.4.

This duality was observed in [13] as an involution of the super algebra
$S(2)$.

######  Example 4.5.

For a split $S(2)$ curve $C$ associated to $C_{0}$ and the vector bundle $E$,
the dual curve $\widehat{C}$ is also split and is associated to $C_{0}$ and
the vector bundle $E^{*}\otimes\Omega_{C_{0}}$.

##  5 Families of super curves

In this section we construct examples of families of supercurves. We construct
families of $S(1|2)$ supercurves that are not $S(2)$ supercurves. Non-split
$S(1|2)$ supercurves over a purely even base. Non-split $S(2)$ supercurves
over a super-scheme.

###  5.1 A family of $S(2)$-super curves

######  Example 5.1.

Consider the $1|2$-dimensional family
$\mathbb{C}^{1|3}\to\mathbb{C}^{0|1}=\mathrm{Spec}(\mathbb{C}[\eta])$, with
$\eta$ an odd variable. Over it we have the two automorphism given by

$\begin{split}A(t|\theta,\rho):=&\ (t+1|\theta,\rho),\\\ B(t|\theta,\rho):=&\
(t+\tau+\theta\eta|\theta,\rho),\end{split}$

with $\tau\in\mathbb{C}$, $\Im(\tau)>0$. The quotient
$\mathbb{T}_{\tau}:=\mathbb{C}^{2|3}\to\mathrm{Spec}(\mathbb{C}[\eta])/\langle
A,B\rangle$ is an analytical family of super torus. In order to see that this
quotient is algebraic, let us recall the Weierstrass function $\wp$ given by
the parameter $\tau$. Then we obtain the closed immersion:

$\begin{split}\mathbb{T}_{\tau}\to&\ \mathbb{P}^{2}(L)\\\
(t|\theta,\rho)\mapsto&\
(\wp(t;\tau+\theta\eta),\partial_{t}\wp(t;\tau+\theta\eta),1|\theta,\rho),\end{split}$

where $E$ is a rank 2 trivial bundle over $\mathbb{P}^{2}$. The image of this
immersion is given by the equation:

$y^{2}=4x^{3}-g_{2}(\tau+\phi\eta)x-g_{3}(\tau+\phi\eta),$ (5.1)

with $(x,y,1|\phi,\psi)\in\mathbb{P}^{2}(L)$. Since (5.1) is even, then
$\mathbb{T}_{\tau}$ is a $1|2$-dimensional family.

To see that this family is not split, suppose that there exists a
$1|0$-dimensional family $M\to\mathrm{Spec}(\mathbb{C}[\eta])$ and a rank 2
bundle $E_{0}$ over the family such that $M(E_{0})=\mathbb{T}_{\tau}$. Observe
that such family should be a family of torus, also the change of coordinates
over any torus should have the form
$\Phi(t|\theta,\rho)=(\phi(t)|\theta\lambda_{11}(t)+\rho\lambda_{12}(t),\theta\lambda_{21}(t)+\rho\lambda_{22}(t))$
and in this case $(\lambda_{ij}(t))$ corresponds to the cocycle of $E_{0}$.
Over $\mathcal{T}_{\mathbb{T}_{\tau}/S}$, $S=\mathrm{Spec}(\mathbb{C}[\eta])$,
we have the global section given by $\partial_{t}$, so we have the exact
sequence

$0\to\langle\partial_{t}\rangle\to\mathcal{T}_{\mathbb{T}_{\tau}/S}\to\mathcal{T}_{\mathbb{T}_{\tau}/S}/\langle\partial_{t}\rangle\to
0.$

Given a change of coordinates
$\Phi(t|\theta,\rho)=(\phi(t)|\theta\lambda_{11}(t)+\rho\lambda_{12}(t),\theta\lambda_{21}(t)+\rho\lambda_{22}(t))$
then the change of coordinates of
$\mathcal{T}_{\mathbb{T}_{\tau}/S}/\langle\partial_{t}\rangle$ are given by
$(\lambda_{ij}(t))$. On the other side, for coordinates $(t,\theta,\rho)$ the
vector field $\partial_{\theta},\partial_{\rho}$ are well defined global
section in $\mathcal{T}_{\mathbb{T}_{\tau}/S}/\langle\partial_{t}\rangle$,
that is $\mathcal{T}_{\mathbb{T}_{\tau}/S}/\langle\partial_{t}\rangle$ is a
trivial bundle, in particular $E_{0}$ is also a trivial bundle. From this, the
tangent bundle over $M(E_{0})$ should be trivial, since the tangent bundle
over the torus is trivial and we can define a global section
$\partial_{\theta^{\prime}}$ for any global section $\theta^{\prime}$ in
$E_{0}$. If this happens, then the space of global sections has dimension
$1|2$. Let us take a section $s$ of $\mathcal{T}_{\mathbb{T}_{\tau}/S}$. With
respect to the etále topology, from the projection
$\mathbb{T}_{\tau}:=\mathbb{C}^{2|2}\to\mathrm{Spec}(\mathbb{C}[\eta])\to\mathbb{T}_{\tau}:=\mathbb{C}^{2|3}\to\mathrm{Spec}(\mathbb{C}[\eta])/\langle
A,B\rangle$, we obtain a section of the tangent bundle
$\mathcal{T}_{\mathbb{C}/S}$, such section should have the form

$\begin{split}s(t|\theta,\rho)=&\ s(t+1|\theta,\rho),\\\ s(t|\theta,\rho)=&\
s(t+\tau+\theta\eta|\theta,\rho).\end{split}$ (5.2)

Using the decomposition
$s(t|\theta,\rho)=a(t|\theta,\rho)\partial_{t}+b(t|\theta,\rho)\partial_{\theta}+c(t|\theta,\rho)\partial_{\rho}$.
From the relations (5.2), we obtain that $b$ should satisfy

$\begin{split}b(t|\theta,\rho)=&\ b(t+1|\theta,\rho),\\\ b(t|\theta,\rho)=&\
b(t+\tau+\theta\eta|\theta,\rho).\end{split}$ (5.3)

from this $b$ should be constant. Analogously $c$ is constant. On the other
side, we get:

$\begin{split}a(t|\theta,\rho)=&\ a(t+1|\theta,\rho),\\\ a(t|\theta,\rho)=&\
a(t+\tau+\theta\eta|\theta,\rho)-b\rho.\end{split}$ (5.4)

Similarly to (5.3), taking derivative on (5.4) we obtain that $a$ is constant
and $b=0$. That is, the vector space of sections has dimension $1|1$, and this
contradicts that such space of sections has dimension $1|2$. Then the family
of torus $\mathbb{T}_{\tau}$ is not split.

This family has a global Berezinian given by $\Delta=[dz|d\theta d\rho]\in
H^{0}(C,\mathrm{Ber}_{C/S})$. Since it has a dual curve given by the quotient
of $\mathbb{C}^{1|3}\to\mathrm{Spec}(\mathbb{C}[\eta])$ by the automorphisms:

$\begin{split}\widetilde{A}(t|\theta,\rho):=&\ (t+1|\theta,\rho),\\\
\widetilde{B}(t|\theta,\rho):=&\ (t+\tau|\theta-\eta,\rho).\end{split}$

The quotient $\widehat{\mathbb{T}}_{\tau}:=\mathbb{C}^{1|3}/\langle
A,B\rangle\to\mathrm{Spec}(\mathbb{C}[\eta])$ is the dual curve. Then
$\mathbb{T}_{\tau}$ is a family of $S(2)$-super curves that is no split.

Also, observe that this family does not have a structure of $SUSY_{2}$-super
curve. Since, if this family has a rank $0|2$ distribution on
$\mathcal{T}_{\mathbb{T}_{\tau}/S}$.

###  5.2 A family of $S(1|2)$-super curves

######  Example 5.2.

Over the affine plane $\mathbb{C}^{1|2}$ consider the following relations

  1. 1.

$A(z|\theta^{1},\theta^{2}):=(z+1|\theta^{1},\theta^{2})$,

  2. 2.

$B(z|\theta^{1},\theta^{2}):=(z+\tau+\theta^{1}\theta^{2}|\theta^{1},\theta^{2})$,

where $\tau$ is even. The quotient $\mathbb{C}^{1|2}/\langle A,B\rangle$ is an
elliptic curve $\mathbb{T}_{\tau}$ with Berezinian
$[dz|d\theta^{1}d\theta^{2}]$. Consider the $SUSY_{4}$-super curve associated,
$\widetilde{\mathbb{T}}_{\tau}$, since
$\mathcal{T}_{\widetilde{\mathbb{T}}_{\tau,\rho}}$ does not have a splitting.
Then $\mathbb{T}_{\tau}$ does not have an $S(2)$-structure.

##  6 On the moduli space of curves with a trivial Berezinian

In [5], Vaintrob studied the moduli space of super curves with a fixed
Berezinian. Given a family $X\to S$ of such supercurves, with special fiber
$X_{0}$ over $s_{0}\in S$, Vaintrob views the generic fiber as a deformation
of the curve $X_{0}$ together with a deformation of a section $\Delta\in
H^{0}(X_{0},\mathrm{Ber}_{X_{0}})$.

We will follow the recipe given in [14] to study the moduli space of
$S(2)$-super curves. If $S$ is a scheme, or a purely-even super-scheme, we
will say that such deformations are _even_. For a super scheme $S$, if we fix
a family $C_{0}\to S_{\mathrm{rd}}$ any extension $C\to S$ is called an _odd_
deformation of $C_{0}\to S_{\mathrm{rd}}$.

We divide the process in three steps:

  1. 1.

We study the restriction of the functor to schemes. Suppose that such functor
is represented by the scheme $A$ and comes with a universal curve
$\mathcal{A}\to A$.

  2. 2.

We focus on odd deformations of such curves. Suppose that such deformation is
given by a functor $S\to E(S)$, where $S$ is a scheme and $E(S)$ is a set.

  3. 3.

We check what kind of automorphisms has our candidate to the moduli space.

The construction is divided in these steps inspired by the following lemmas:

######  Lemma 6.1.

Let $M$ be a super scheme, that defines the functor $h_{M}:SSch\to Sets$,
$h_{M}(N)=Hom_{SSch}(N,M)$, then the restriction $h_{M}|_{Sch}$ is represented
by $M_{\mathrm{rd}}$.

###### Proof.

This follows directly from Lemma 2.6. ∎

######  Lemma 6.2.

Let $M$ be a split super scheme, defined by the scheme
$j:M_{\mathrm{rd}}\hookrightarrow M$ and a locally free
$\mathcal{O}_{M_{\mathrm{rd}}}$-module $E$, then
$E^{*}\simeq\ker(j^{*}\Omega_{M}\to\Omega_{M_{\mathrm{rd}}})$.

###### Proof.

This follows directly from Observation 3.5. ∎

That is, if our moduli space is given by a locally split super scheme, then
such space should be described by the reduced space representing even
deformations and the fiber bundle defined (locally) by the purely odd
deformations with a fixed even base. To finish this process, we find the
automorphisms that fix such families of curves.

######  Observation 6.3.

The process described above is justified by the following construction:
Suppose that we have a family $X\to S$ of $S(2)$ curves, then the diagram

$\textstyle{X_{0}:=X\times_{S_{\mathrm{rd}}}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{\mathrm{rd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$
(6.1)

Since $S_{\mathrm{rd}}$ is a scheme and $X_{0}\to S_{\mathrm{rd}}$ is family
of $S(2)$ super curves, then this family is given by a morphism
$\phi_{0}:S_{\mathrm{rd}}\to A$ and the pullback $\phi_{0}^{*}\mathcal{A}\to
S_{\mathrm{rd}}$. Since locally, $S$ is a split scheme, then in an open
neighbourhood $U\subset S_{\mathrm{rd}}$ there exists a fiber bundle $E$ such
that, locally, $S|_{U}=U(E)$. Assuming that $U$ is affine we will proof
uniqueness on the extension of such family $X_{0}(U)\to U$.

Finally the family $\pi:X\to S$ is going to be described as the gluing of
local open pieces $\\{\pi^{-1}(U_{i})\to U_{i}\\}_{i}$ through morphisms
$U_{i}\to A$. Finally, the inner automorphism is going to give us an orbifold
description of such object.

###  6.1 The even part

Observe that for an $S(2)$-super curve, over its $SUSY_{4}$-super curve
$\widetilde{C}$ we have the local vector fields
$D^{i}_{+}=\partial_{\rho^{i}}$ and
$D^{j}_{-}=\rho^{j}\partial_{z}+\partial_{\theta^{j}}$ on the coordinate
charts $(z|\theta^{1},\theta^{2},\rho^{1},\rho^{2})$ given by (3.21)
satisfying the relation $\\{D^{i}_{+},D^{j}_{-}\\}=\delta_{ij}\partial_{z}$.
The change of coordinates satisfies equations (3.15):

$D^{j}_{-}F=\eta^{1}D^{j}_{-}\phi^{1}+\eta^{2}D^{j}_{-}\phi^{2},$

thus we have

$\\{D^{i}_{+},D^{j}_{-}\\}F=D^{i}_{+}\eta^{1}D^{j}_{-}\phi^{1}-\eta^{1}\\{D^{i}_{+},D^{j}_{-}\\}\phi^{1}+D^{i}_{+}\eta^{2}D^{j}_{-}\phi^{2}-\eta^{2}\\{D^{i}_{+},D^{j}_{-}\\}\phi^{2},$

and that implies:

$\delta_{ij}(\partial_{z}F+\eta^{1}\partial_{z}\phi^{1}+\eta^{2}\partial_{z}\phi^{2})=D^{i}_{+}\eta^{1}D^{j}_{-}\phi^{1}+D^{i}_{+}\eta^{2}D^{j}_{-}\phi^{2}.$

It follows from (3.18) that

$\left[\begin{matrix}D^{1}_{+}\eta^{1}&D^{1}_{+}\eta^{2}\\\
D^{2}_{+}\eta^{1}&D^{2}_{+}\eta^{2}\end{matrix}\right]=\left[\begin{matrix}D^{2}_{-}\phi^{2}&-D^{2}_{-}\phi^{1}\\\
-D^{1}_{-}\phi^{2}&D^{1}_{-}\phi^{1}\end{matrix}\right].$

Finally, we get

$\partial_{z}F+\eta^{1}\partial_{z}\phi^{1}+\eta^{2}\partial_{z}\phi^{2}=D^{1}_{-}\phi^{1}D^{2}_{-}\phi^{2}-D^{1}_{-}\phi^{2}D^{2}_{-}\phi^{1},$
(6.2)

then we have that the projection:

$\det E\to\Omega_{C},$

is an isomorphism.

Conversely, suppose that we start with a $1|0$-family over a purely even base
$S$, $\pi_{0}:C_{0}\to S$, and a rank $2|0$ bundle $E$ with an isomorphism
$\beta:\det E\to\Omega^{1}_{C_{0}}$. Given such data, we construct $\pi:C\to
S$ given by

$\begin{split}C_{\text{rd}}=&\ C_{0},\\\ \mathcal{O}_{C,0}=&\
\mathcal{O}_{C_{0}}\oplus\det E,\\\ \mathcal{O}_{C,1}=&\ \Pi E.\end{split}$

The $S(2)$ structure is given (locally) by the coordinates $z$ and a local
frame $\theta^{1},\theta^{2}$ of $E$ such that
$\beta(\theta^{1}\otimes\theta^{2})=dz$. This data defines the local section
of the Berezinian $[dz|d\theta^{1}d\theta^{2}]$. Since $\beta$ is an
isomorphism, then such class is well defined. Also, observe that for any
change of coordinates compatible with the Berezinian,
$\Phi=(F|\phi^{1},\phi^{2})$, where $F(z|\theta^{1},\theta^{2})=F(z)$, the
decomposition (in Proposition 2.22), $\Phi=\exp(X)\circ\phi$, gives us that
$X\in S(2)$.

Similar to [2] we obtain:

######  Proposition 6.4.

For any family $\pi_{0}:C_{0}\to S$ of relative dimension $1|0$ over a purely
even base, the following data are equivalent:

  1. 1.

An $S(2)$-family of curves $\pi:C\to S$ over $S$ with
$C_{\text{rd}}=C_{0,\text{rd}}$ and
$\mathcal{O}_{C,0}=\mathcal{O}_{C_{0}}(1)$, where $C_{0}(1)$ is the first
neighborhood of the diagonal in $C_{0}\times_{S}C_{0}$.

  2. 2.

A rank $2$ bundle $E$ and an isomorphism $\det
E\overset{\beta}{\to}\Omega_{C_{0}/S}$ up to equivalence: a pair $(E,\beta)$
is equivalent to $(E^{\prime},\beta^{\prime})$ is there exists an isomorphism,
such that the following diagram commutes

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.75693pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\det
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.75693pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.57336pt\raise-25.65274pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
44.67123pt\raise-32.17424pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
38.75693pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\det
E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
54.28384pt\raise-19.54164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\beta^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
54.28384pt\raise-29.24998pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
44.67123pt\raise-39.0833pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega_{C_{0}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

###### Proof.

The previous comment shows $(2)\to(1)$.

To see $(1)\to(2)$ consider the $SUSY_{4}$-super curve $\widetilde{C}$
associated to $C$. Since $C$ is $S(2)$, we have the well defined $0|2$ bundle

$\widehat{E}=\langle\rho^{1}\partial_{z}+\partial_{1},\rho^{2}\partial_{z}+\partial_{2}\rangle$

over $\widetilde{C}$. The pullback $E:=i^{*}(\Pi\widehat{E})$ over the
inclusion $i:C_{\text{rd}}\hookrightarrow\widetilde{C}$ is a rank $2$ bundle.
We get from equation (6.2) that $\det E\simeq\Omega_{C_{0}}$, where
$C_{0}=C_{\text{rd}}$. Finally, since $C$ is $S(2)$, then
$\mathcal{O}_{C,0}=\mathcal{O}_{C_{0}}\oplus\det E$ and $\mathcal{O}_{C,1}=\Pi
E$. ∎

In this case, we have that such moduli space is parametrized by the space of
curves, $C_{0}$, and a rank 2 fibre bundles $E$ with a marked isomorphism
$\det E\simeq\Omega_{C_{0}}$.

######  Observation 6.5.

To calculate the dimension of the moduli space over a point $[(C,\Delta)]$
consider a deformation given by the class $\\{X_{ij}\\}\in
H^{1}(C,\mathcal{T}_{C})$ with respect to a covering $\\{U_{i}\\}_{i}$. If we
want that the perturbation keep in the space of curves with a trivial
Berezinian then we need that the class of
$\\{\mathrm{sdiv}_{\Delta}(X_{ij})\\}\in H^{1}(C,\mathcal{O}_{C})$ to be zero.
So, the space of variations of curves with a trivial Berezinian is given by
the kernel of the homomorphism of group

$\begin{split}H^{1}(C,\mathcal{T}_{C})\stackrel{{\scriptstyle\phi}}{{\to}}&\
H^{1}(C,\mathcal{O}_{C})\\\ \\{X_{ij}\\}\mapsto&\
\\{\mathrm{sdiv}_{\Delta}(X_{ij})\\}\end{split}$

Similarly, if we want deformations of $S(2)$-super curves also we need that
such perturbation is in the kernel of

$\begin{split}\ker(\phi)\stackrel{{\scriptstyle\psi}}{{\to}}&\
H^{1}\left(C,\mathrm{Aut}^{\Delta}[[1|2]]\backslash\mathrm{Aut}^{\delta}_{C}\right)\\\
A=\\{X_{ij}\\}\mapsto&\ \Gamma_{A},\end{split}$

recall that $\Gamma_{A}=\\{\exp(X_{ij})\\}_{ij}$.

Finally, the dimension of the moduli space of $S(2)$-super curves over
$[\mathcal{C}]$ is given by $\ker(\psi)$ and
$H^{0}(\mathcal{C},\text{Ber}_{\mathcal{C}})$.

Given a genus $g$, and a degree $d=g-1$. For the moduli space of genus $g$
curves with a rank 2 vector bundle with a fixed degree $d$, denoted by
$\mathcal{M}_{g,2,d}$ we consider the morphism to the space of curves with a
line bundle with a fixed degree $2d$, $P_{g,2d}$, given by:

$\begin{split}\mathcal{M}_{g,2,d}\to&P_{g,2d}\\\ [E\to C]\mapsto&[\det E\to
C].\end{split}$

We obtain that the preimage of $\\{[\Omega_{C}\to C]\\}$, denoted by
$M_{g,2,\Omega}$, is the space of curves with genus $g$ and a rank 2 bundle
$E$ together with an isomorphism $\det E\to\Omega_{C}$. Recall that the smooth
part is given by the stable bundles.

The universal curve is given by the projection $X(E)\mapsto[E\to X]$.

######  Observation 6.6.

The involution described in Section 4 is exactly the same as the one mentioned
in Observation 3.27.

###  6.2 The odd part

Let $\pi:C\to S$ be a family of $S(2)$super curves with a fixed genus $g$, for
a split supermanifold $S$, with reduced space $S_{\mathrm{rd}}$ and
$\mathcal{O}_{S_{\mathrm{rd}}}$-free module $W$. Considering the closed point
$s_{0}\in S$ and the fiber $C_{s_{0}}\to\\{s_{0}\\}$, there exists a covering
by affine open sets $\\{U_{i}\\}$ of $C_{s_{0}}$ such that the family $C\to S$
restricts to $U_{i}\times S\to S$. This observation follows from [15, Theorem
1.2.4]. Then the global family is determined by cocycle given by the change of
coordinates in $\Phi_{ij}\in\mathrm{Aut}((U_{i}\cap U_{j})\times S)$.

Considering the reduced family $C_{0}\to S_{\mathrm{rd}}$ given by (6.1),
fixing the reduced family, then the reduced part of the change of coordinates
$\Phi_{ij}$ is fixed, denote it by $\phi_{ij}\in\mathrm{Aut}((U_{i}\cap
U_{j})\times S)_{\mathrm{rd}}$. Then, the $\phi_{ij}^{-1}\Phi_{ij}$ is an
automorphism being the identity over the reduced space, then this morphism is
the exponential of a nilpotent vector field relative to $S$. To get a family
of $S(2)$ curves we need the distribution:

$\mathcal{D}:=S(2)\cap\mathcal{T}_{C/S}$

where $S(2)$ was described in (3.1), and the sheaf $\mathcal{N}$ of nilpotent
elements of $\Lambda^{\bullet}\pi^{*}W$. Then,
$\phi_{ij}^{-1}\Phi_{ij}=\exp(X_{ij})$ for an even vector field
$X_{ij}\in\mathcal{N}\otimes\mathcal{D}$. The group bundle

$G=\exp((\mathcal{N}\otimes\mathcal{D})_{0}),$ (6.3)

and the deformations are parametrized by $H^{1}(C_{0},G)$.

Finally, we get

######  Proposition 6.7.

Let $S$ be a split super scheme and a family $C\to S$ of $S(2)$-super curves,
then this family correspond to a extension of $C_{0}\to S_{\mathrm{rd}}$ and a
class in $H^{1}(C_{0},G)$, for $G$ in (6.3).

###  6.3 Inner Automorphism

Now, we will check which automorphisms preserves the family of curves. We are
looking for maps $\psi:S\to S$ such that the families of $S(2)$ super curves
$C\to S$, $\psi^{*}C\to S$ are equal, in this case we are going to say that
$\psi:S\to S$ preserve the family. First consider the following lemma:

######  Lemma 6.8.

Let $\pi:C\to S$ be a family of curves, suppose that $\psi:S\to S$ preserve
the family and $\psi_{\mathrm{rd}}:S_{\mathrm{rd}}\to S_{\mathrm{rd}}$ is the
identity, then $\psi$ is the identity.

###### Proof.

Since, we can cover $S$ by open split schemes $\\{S_{i}\\}_{i}$, then
$\psi|_{C_{i}}:C_{i}\to S_{i}$, with $C_{i}=\pi^{-1}(S_{i})$, is the identity
over $S_{i,\mathrm{rd}}$. The family $C_{i}\to S_{i}$ is defined by
$C_{i,0}\to S_{i,\mathrm{rd}}$ and a class $H^{1}(C_{i,0},G_{i})$. Since
$\psi$ define the same family, then both classes should coincide, then $\psi$
should be also the identity. ∎

From the previous lemma, it follows that any automorphism of the family $X\to
S$ is given by an automorphism over the reduced space $S_{\mathrm{rd}}$.

Let $\pi:C\to S$ be a family of $S(2)$-super curves, $S$ is a scheme and $E$
be the rank 2 bundle defining $\pi$. Considering the reduction
$C_{\mathrm{rd}}\to S$, we obtain a family of genus $g$ curves over $S$. From
now on we consider only the case $g\geq 2$. There does not exists any
automorphism different from the identity. Then, in order to study such
automorphism we have to check what happens in the odd part generated by the
bundle $E$, that is we have to study the automorphisms of the rank $2$ bundle
$E\to C_{\mathrm{rd}}$, with the chosen isomorphism $\det E\to
C_{\mathrm{rd}}$. Considering the stable bundles, we have
$\mathrm{Aut}_{C_{\mathrm{rd}}}(E)=\pi^{*}\mathcal{O}_{S}^{*}$, since we need
that such automorphism preserve the isomorphism $\det E\to C_{\mathrm{rd}}$,
this group reduces to $\pm 1$. The corresponding automorphism given by $1$ is
the identity, while the automorphism given by $-1$ is denoted by $\Psi$. Then
we get the lemma:

######  Lemma 6.9.

Let $\pi:C\to S$ be a family of genus $g$ $S(2)$-super curves, $S$ is a scheme
and $E$ be the rank 2 stable bundle defining $\pi$. Then any automorphism
$\psi:S\to S$ preserving the family corresponds to a class $\sigma_{\psi}\in
H^{1}(S,\mathbb{Z}/2\mathbb{Z})$.

Finally, we get

######  Proposition 6.10.

Let $\pi:C\to S$ be a family of genus $g$ $S(2)$-super curves. Then any
automorphism $\psi:S\to S$ preserving the family corresponds to a class
$\sigma_{\psi}\in H^{1}(S_{\mathrm{rd}},\mathbb{Z}/2\mathbb{Z})$.

######  Theorem 6.11.

The data $(M_{g,2,\Omega},\\{U_{i}\\}_{i},\delta_{ij})$, where
$M_{g,2,\Omega}$ is the moduli space of genus $g$ curves $C$ joint with a rank
2 bundle $E\to C$ with a fixed isomorphism $\det E\simeq\Omega_{C}$, $U_{i}$
is a split super scheme such that $\cup U_{i,\mathrm{rd}}=M_{g,2,\Omega}$ and
the associated bundle is given in Proposition 6.7 considering the universal
family $C_{U_{i,\mathrm{rd}}}\to U_{i,\mathrm{rd}}$, and
$\delta_{ij}:U_{i}|_{U_{ij,\mathrm{rd}}}\to U_{j}|_{U_{ij,\mathrm{rd}}}$
isomorphism such that
$\delta_{ij}\circ\delta_{jk}\circ\delta_{ki}\in\\{\text{id},\Psi\\}$; models
the orbifold representing the functor of $S(2)$ super curves.

###  6.4 The even data for $S(1|2)$-super curves

A general family of $S(1|2)$-super curve $C\to S$ over the even base $S$, with
a nonvanishing section $\Delta\in H^{0}(C,\mathrm{Ber}_{C/S})$, defines a
class

$\Gamma_{C}\in H^{1}(C,\pi^{*}\mathcal{O}_{S}),$ (6.4)

given by the bundle $K$ in (3.3). Suppose that this class is defined by a
covering $\\{U_{i}\\}$ with local coordinates $\Phi_{i}$ compatible to
$\Delta$ and $\Gamma_{C}=\\{\lambda_{ij}\\}$, then the change of coordinates
is given by

$\begin{split}z_{j}=&\
F_{ij}(z_{i})+\lambda_{ij}\partial_{z_{i}}F_{ij}(z_{i})\theta^{1}_{i}\theta^{2}_{i}\\\
\theta^{1}_{j}=&\ \theta^{1}_{i}a_{11}(z_{i})+\theta^{2}_{i}a_{12}(z_{i})\\\
\theta^{2}_{j}=&\
\theta^{1}_{i}a_{21}(z_{i})+\theta^{2}_{i}a_{22}(z_{i}).\end{split}$

Here the covering $\\{U_{i}\\}$ and the change of coordinates $\\{F_{ij}\\}$
defines a curve $C_{0}$ and

$A_{ij}=\left[\begin{matrix}a_{11}(z_{i})&a_{12}(z_{i})\\\
a_{21}(z_{i})&a_{22}(z_{i})\end{matrix}\right]$

defines a rank 2 bundle $E$ over $C_{0}$ with $\det E\simeq\omega_{C_{0}}$. We
can define the new curve $C^{\prime}=C_{0}(E)$ that is actually an
$S(2)$-super curve.

The sheaf $\mathcal{O}_{C,0}$ defined over $C_{0}$ is a sheaf of algebras that
is isomorphic to $C_{0}(1)$, the first neighbourhood of the diagonal on
$C_{0}\times_{S}C_{0}$, if and only if $\\{\lambda_{ij}\\}\in
H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$ vanishes. In general, each class
$\Gamma\in H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$ defines a curve
$C_{\Gamma}=(C_{0},\mathcal{O}_{C,0})$ by the diagram from (3.8). First take
$\mathcal{L}=L\otimes\mathcal{O}_{C_{0}}$ and
$\mathcal{A}=\alpha(\mathcal{O}_{C_{0}})$, with these we construct the sheaf

$\begin{split}\mathcal{O}_{C,0}:=\frac{T^{\bullet\geq 1}\mathcal{L}}{\langle
x\otimes y-y\otimes x,a\otimes
a:x,y\in\mathcal{L},a\in\mathcal{A}\rangle}.\end{split}$

Observe that this sheaf fits in the square-zero extension of algebras:

$\begin{split}0\to\Omega_{C_{0}}\to\mathcal{O}_{C,0}\to\mathcal{O}_{C_{0}}\to
0,\end{split}$

Since we consider terms of degree greater then zero, then $\mathcal{O}_{C,0}$
does not necessarily have the structure of an $O_{C_{0}}$-algebra.

Finally, we consider

$\begin{split}\mathcal{O}_{C,1}:=\Pi E,\end{split}$

with multiplication given by the isomorphism

$\begin{split}\det E\to\Omega_{C_{0}}.\end{split}$

So, we get the $S(2)$-super curve $(C,\mathcal{O}_{C})$. Finally:

######  Proposition 6.12.

For any family over a purely even base $\pi_{0}:C_{0}\to S$ of relative
dimension $1|0$, the following data are equivalent:

  1. 1.

An $S(1|2)$-family of curves $\pi:C\to S$ over $S$ with
$C_{\text{rd}}=C_{0,\text{rd}}$ and
$\mathcal{O}_{C,0}=\mathcal{O}_{C_{\Gamma}}$.

  2. 2.

A class $\Gamma\in H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$ and a rank $(2|0)$
bundle $E$ with an isomorphism $\det E\overset{\beta}{\to}\Omega_{C_{0}/S}$ up
to equivalence: a triple $(E,\beta,\Gamma)$ is equivalent to
$(E^{\prime},\beta^{\prime},\Gamma^{\prime})$ if there exists an isomorphism
$\alpha:E\rightarrow E^{\prime}$, such that $\Gamma=\Gamma^{\prime}$ and the
diagram

$\textstyle{\det
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\det
E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{\prime}}$$\textstyle{\Omega_{C_{0}/S}}$

commutes.

###### Proof.

The previous comment shows $(2)\to(1)$.

To see $(1)\to(2)$ we get the class $\Gamma\in
H^{1}(C_{0},\pi^{*}\mathcal{O}_{S})$ by taking
$\lambda_{ij}=\gamma(\Phi_{ij})$ as in (6.4), and considering the $S(2)$-super
curve $C^{\prime}=C(E)$, from the comment, the associated $C^{\prime}$ super
curve verifies what we want. ∎

###  6.5 The genus 1 curve

For the special case of genus 1 curves, we have that the even part is given by
an ordinary curve $E_{0}$ and an element of $H^{1}(E_{0},SL(2,\mathbb{C}))$.

Is known that the space of elliptic curves is given by a quotient of the upper
half space $\mathbb{H}$ by the group $SL(2,\mathbb{Z})$, where any element
$\tau\in\mathbb{H}$ defines a quotient of $\mathbb{C}$ by the action of the
group $\mathbb{Z}_{\tau}=\\{a+b\tau:a,b\in\mathbb{Z}\\}$. Similarly, any
element $(\tau,a)\in\mathbb{H}\times\mathfrak{sl}(2,\mathbb{C})$, the quotient
of $\mathbb{C}^{1|2}$ by the action of
$A(z|\theta^{1},\theta^{2})=(z+1|\theta^{1},\theta^{2})$ and
$B(z|\theta^{1},\theta^{2})=(z+\tau|(\theta^{1},\theta^{2})\exp(2\pi ia))$.

A family of even deformation is given by
$\mathbb{H}\times\mathfrak{sl}(2,\mathbb{C})$ considering the action of the
following three groups: $SL(2,\mathbb{Z})$, $\mathbb{Z}^{2}$,
$SL(2,\mathbb{C})$.

  1. 1.

The group $SL(2,\mathbb{Z})$: Consider the action

$\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\cdot(\tau,a)=\left(\frac{a\tau+b}{c\tau+d},\frac{a}{c\tau+d}\right)$

and for the quotient, we have the isomorphism induced by

$\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\cdot(z|\theta^{1},\theta^{2})=\left(\frac{z}{c\tau+d}\Bigg{|}(\theta^{1},\theta^{2})\exp\left(-2\pi
iz\frac{ca}{c\tau+d}\right)\right)$

and observe that such isomorphism preserves the Berezinian if and only if
$c=0$ and $d=1$.

  2. 2.

The group $\mathbb{Z}^{2}$: Consider the action

$(m,n)\cdot(\tau,a)=(\tau,a+m\tau+n)$

and for the quotient, we have the isomorphism induced by

$(m,n)\cdot(z|\theta^{1},\theta^{2})=(z|(\theta^{1},\theta^{2})\exp\left(2m\pi
iz\right))$

and observe that such isomorphism preserves the Berezinian if and only if
$m=0$.

  3. 3.

The group $SL(2,\mathbb{C})$: Consider the action

$C\cdot(\tau,a)=(\tau,CaC^{-1})$

and for the quotient, we have the isomorphism induced by

$C\cdot(z|\theta^{1},\theta^{2})=(z|(\theta^{1},\theta^{2})C)$

and observe that such isomorphism preserves the Berezinian.

Finally, if we consider the group $SL(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}\times
SL(2,\mathbb{C})$ with the product

$(A,\gamma,C)\cdot(A^{\prime},\gamma^{\prime},C^{\prime})=(AA^{\prime},\gamma^{\prime}+\gamma
A^{\prime},CC^{\prime})$

we get the fine moduli space
$\mathcal{M}_{0}=\mathbb{H}\times\mathfrak{sl}(2,\mathbb{C})\sslash
SL(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}\times SL(2,\mathbb{C})$ of the even
families of $S(2)$-super curves.

######  Observation 6.13.

The moduli space of orientable $SUSY_{2}$-super curves is given by $(\tau,a)$,
where $a\in\mathfrak{sl}(2,\mathbb{C})$ is diagonal. In this case and
coordinates given by $(z|\theta^{1},\theta^{2})$, the $SUSY$-structure is
obtained by $\omega=dz+\theta^{2}d\theta^{1}$.

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