# Extended clutching construction for the moduli of stable curves

Alexander Polishchuk  Department of Mathematics, University of Oregon, Eugene,
OR 97403, USA; National Research University Higher School of Economics; and
Korea Institute for Advanced Study

###### Abstract.

We give a description of the formal neighborhoods of the components of the
boundary divisor in the Deligne-Mumford moduli stack
$\overline{{\mathcal{M}}}_{g}$ of stable curves in terms of the extended
clutching construction that we define. This construction can be viewed as a
formal version of the analytic plumbing construction. The advantage of our
formal construction is that we can control the effect of changing formal
parameters at the marked points that are being glued. As an application, we
prove that the 1st infinitesimal neighborhood of the boundary component
$\Delta_{1,g-1}$ is canonically isomorphic to the 1st infinitesimal
neighborhood in the normal bundle, near the locus
${\mathcal{M}}_{1,1}\times{\mathcal{M}}_{g-1,1}\subset\Delta_{1,g-1}$
corresponding to pairs of smooth curves with marked points. As another
application, we show how to study the period map near the boundary components
$\Delta_{g_{1},g_{2}}$ in terms of the coordinates coming from our extended
clutching construction.

Partially supported by the NSF grant DMS-2001224, and within the framework of
the HSE University Basic Research Program and by the Russian Academic
Excellence Project ‘5-100’.

##  1\. Introduction

Let $\overline{{\mathcal{M}}}_{g}$ be the moduli stack of stable curves of
genus $g\geq 2$. Consider the component
$\Delta_{g_{1},g_{2}}\subset\overline{{\mathcal{M}}}_{g}$ of the boundary
divisor corresponding to reducible curves with components of genus $g_{1}$ and
$g_{2}$, where $g=g_{1}+g_{2}$. We are interested in a description of the
$n$th order thickening $\Delta_{g_{1},g_{2}}^{(n)}$ of $\Delta_{g_{1},g_{2}}$
in $\overline{{\mathcal{M}}}_{g}$ for any $n\geq 1$.

Recall that there is a natural “clutching” surjective finite morphism

$\xi_{g_{1},g_{2}}:\overline{{\mathcal{M}}}_{g_{1},1}\times\overline{{\mathcal{M}}}_{g,2,1}\to\Delta_{g_{1},g_{2}}$

which for $g_{1}\neq g_{2}$ is generically an isomorphism (see [1, ch. XII,
Prop. 10.11]).

For every $n\geq 1$, let us set
$S_{n}=\operatorname{Spec}(\operatorname{\mathbb{Z}}[q]/(q^{n+1}))$. Let
$\overline{{\mathcal{M}}}^{(\infty)}_{g,1}\to\overline{{\mathcal{M}}}_{g,1}$
denote the moduli space of stable curves of genus $g$ with one marked point
together with a formal parameter at the marked point.

We show that the clutching morphism extends naturally to a collection of
compatible morphisms

$\xi_{g_{1},g_{2}}^{(n)}:\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},1}\times\overline{{\mathcal{M}}}^{(\infty)}_{g_{2},1}\times
S_{n}\to\Delta_{g_{1},g_{2}}^{(n)},$

for $n\geq 0$. The corresponding family of stable curves is defined by gluing
the constant deformations of the complements to the marked points with the
standard deformation of the node $x_{1}x_{2}=q$. This can be viewed as an
algebraic analog of the plumbing construction, well known in the Teichmüller
space approach to the moduli of curves (see [3]).

The price one pays for getting these extended morphisms is the need to fix
formal parameters to be glued. However, we show that these choices can be
controlled in a way compatible with the morphisms $\xi_{g_{1},g_{2}}^{(n)}$.
The setup for this is the well known action of the Lie algebra of vector
fields on the moduli spaces $\overline{{\mathcal{M}}}^{(\infty)}_{g,1}$, or
more precisely of the ind-group ${\mathcal{G}}$ whose $R$-points are
continuous automorphisms of $R(\\!(t)\\!)$ (see [5], [2, Sec. 17.3]).

We construct a group scheme ${\mathcal{G}}_{{\operatorname{glue}},n}$ over
$S_{n}$ together with an embedding

${\mathcal{G}}_{{\operatorname{glue}},n}\hookrightarrow({\mathcal{G}}\times{\mathcal{G}})_{S_{n}},$

so that we can consider the induced action of
${\mathcal{G}}_{{\operatorname{glue}},n}$ on

$\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2},n}:=\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},1}\times\overline{{\mathcal{M}}}^{(\infty)}_{g_{2},1}\times
S_{n}.$

Our main result (see Theorem 1.1 below) states that the extended clutching
morphism $\xi_{g_{1},g_{2}}^{(n)}$ factors through the quotient by
${\mathcal{G}}_{{\operatorname{glue}},n}$ and that the map from this quotient
to the $n$th neighborhood of $\Delta_{g_{1},g_{2}}$ is generically a
${\mathbb{G}}_{m}$-torsor for $g_{1}\neq g_{2}$ (there is an extra double
covering in the case $g_{1}=g_{2}$).

The same construction works for the component
$\Delta_{0}\subset\overline{{\mathcal{M}}}_{g}$ of the boundary divisor
corresponding to curves with a non-separating node. Namely, the clutching
morphism

$\xi_{0}:\overline{{\mathcal{M}}}_{g-1,2}\to\Delta_{0}$

extends to a collection of morphisms

$\xi_{0}^{(n)}:\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}\times
S_{n}\to\Delta_{0}^{(n)},$

where we consider the moduli space with choices of parameters at both marked
points. We still have the action of ${\mathcal{G}}_{{\operatorname{glue}},n}$
on $\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}$, so that $\xi_{0}^{(n)}$
factors through the quotient.

To formulate the statement more precisely let us introduce some notation. Let
${\mathcal{M}}^{(\infty)}_{g_{1},g_{2},n}\subset\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2},n}$
be the open locus corresponding to pairs of smooth curves, and let
$U\Delta_{g_{1},g_{2}}^{(n)}\subset\Delta_{g_{1},g_{2}}^{(n)}$ be the
corresponding open locus in $\Delta_{g_{1},g_{2}}^{(n)}$ corresponding to
curves with only one node. In the case $g_{1}=g_{2}=g/2$ we consider the étale
double coverings

${\mathcal{E}}_{g/2,g/2}\to U\Delta_{g/2,g/2},\ \
{\mathcal{E}}_{g/2,g/2}^{(n)}\to U\Delta_{g/2,g/2}^{(n)}$

corresponding to a choice of one of the components of the stable curve with a
separating node. Similarly, we consider the open locus $U\Delta_{0}^{(n)}$ of
curves with only one node and the étale double coverings

${\mathcal{E}}_{0}\to U\Delta_{0},\ \ {\mathcal{E}}_{0}^{(n)}\to
U\Delta_{0}^{(n)}$

corresponding to ordering the preimages of the node in the normalization. For
$g_{1}\neq g_{2}$ we set
${\mathcal{E}}_{g_{1},g_{2}}^{(n)}=U\Delta_{g_{1},g_{2}}^{(n)}$. Note that in
the terminology of [1] these stacks correspond to choices of a
$\Gamma$-structure, where $\Gamma$ is the graph with two vertices marked by
$g_{1}$ and $g_{2}$ and an edge between them, in the case of
${\mathcal{E}}_{g_{1},g_{2}}$, and $\Gamma$ is the graph with the single
vertex and a loop in the case of ${\mathcal{E}}_{0}$.

######  Theorem 1.1.

The morphism $\xi_{g_{1},g_{2}}^{(n)}$ (resp., $\xi_{0}^{(n)}$) factors
through a morphism

$\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2},n}/{\mathcal{G}}_{{\operatorname{glue}},n}\to\Delta_{g_{1},g_{2}}^{(n)}$

(resp.,
$\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}/{\mathcal{G}}_{{\operatorname{glue}},n}\to\Delta_{0}^{(n)}$).
The restriction of this morphism to the locus of smooth curves,

${\mathcal{M}}^{(\infty)}_{g_{1},g_{2},n}/{\mathcal{G}}_{{\operatorname{glue}},n}\to
U\Delta_{g_{1},g_{2}}^{(n)}$

(resp.,
${\mathcal{M}}^{(\infty)}_{g-1,2}//{\mathcal{G}}_{{\operatorname{glue}},n}\to
U\Delta_{0}^{(n)}$) is canonically identified with the
${\mathbb{G}}_{m}$-torsor over ${\mathcal{E}}_{g_{1},g_{2}}^{(n)}$
corresponding to the pull-back of ${\mathcal{O}}(\Delta)$ on
$\overline{{\mathcal{M}}}_{g}$.

As an application, in the case $g_{1}=1$ we can identify the 1st infinitesimal
neighborhood of the boundary divisor with the one in the normal line bundle,
at least over the smooth locus of the divisor. In the case $g_{1}=g_{2}=1$, we
can do the same for any $n$th infinitesimal neighborhood. Let us denote by
${\mathcal{N}}$ the normal bundle to the embedding of the boundary divisor
into $\overline{{\mathcal{M}}}_{g}$.

######  Theorem 1.2.

Let us work over $\operatorname{Spec}\operatorname{\mathbb{Q}}$.

(i) For any $g\geq 3$, there is a natural isomorphism of
$U\Delta_{1,g-1}^{(1)}$ with the $1$st infinitesimal neighborhood of the zero
section in the normal bundle to $U\Delta_{1,g-1}$.

(ii) Now consider the case of $g=2$ and any $n\geq 0$. Let us denote by
${\mathcal{N}}_{{\mathcal{E}}}$ the pull-back of the normal bundle to
${\mathcal{E}}_{1,1}\to U\Delta_{1,1}$ (resp., ${\mathcal{E}}_{0}\to
U\Delta_{0}$). Then there is a natural isomorphism of
${\mathcal{E}}_{1,1}^{(n)}$ (resp., ${\mathcal{E}}_{0}^{(n)}$) with the $n$th
infinitesimal neighborhood of the zero section in the line bundle
${\mathcal{N}}_{{\mathcal{E}}}$ over ${\mathcal{E}}_{1,1}$ (resp., over
${\mathcal{E}}_{0}$).

The extended clutching morphisms provide convenient “gluing” coordinates in
the formal neighborhood of the components of the boundary divisor in
$\overline{{\mathcal{M}}}_{g}$. To illustrate this we show how to compute the
expansion in powers of $q$ (the gluing coordinate), defined on the formal
neighborhood of $\Delta_{g_{1},g_{2}}$, of the morphism

$\pi_{*}\omega_{C/{\mathcal{M}}_{g}}\to
R^{1}\pi_{*}\operatorname{\mathbb{C}}_{C/{\mathcal{M}}_{g}}$

used to define the period map (see Section 5). Here
$\pi:C\to{\mathcal{M}}_{g}$ is the universal curve. In particular, we get
explicit formulas for the expansion modulo $q^{g_{1}+g_{2}+2}$, and in the
case $g_{1}=g_{2}=1$, a recursive procedure for computing the complete
$q$-expansion.

The results of this paper admit counterparts for the moduli spaces of stable
supercurves that will be treated elsewhere. In the super case the “gluing
coordinates” near the components of the boundary divisors are useful for
studying the polar behavior of the superanalog of the Mumford’s isomorphism
and of the superstring supermeasure.

The paper is organized as follows. In Sec. 2 we define and study the group
scheme ${\mathcal{G}}_{{\operatorname{glue}},n}$ which acts on deformations of
the node singularity. In Sec. 3 we define the extended clutching construction.
In Sec. 4 we prove Theorems 1.1 and 1.2. Finally, in Sec. 5 we study the
period map near the boundary divisor $\Delta_{g_{1},g_{2}}$ using the
coordinates given by the extended clutching construction.

Acknowledgment. I am grateful to Giovanni Felder and David Kazhdan for
discussions of the analogous picture for supercurves, which led me to consider
the classical case.

##  2\. Changes of coordinates on a node deformation

###  2.1. The gluing group scheme

For any commutative ring $R$ let us denote by $\operatorname{Aut}R[\\![t]\\!]$
the group of continuous automorphisms of $\operatorname{Aut}R[\\![t]\\!]$,
identical on $R$. Similarly we denote by $\operatorname{Aut}R(\\!(t)\\!)$ the
group of continuous automorphisms of $R(\\!(t)\\!)$, identical on $R$. The Lie
algebra of this group functor is ${\mathcal{W}}$, the Witt algebra over
$\operatorname{\mathbb{Z}}$. It has a topological basis
$L_{n}=z^{n+1}\frac{d}{dz}$, $n\in\operatorname{\mathbb{Z}}$, and the bracket
is given by

$[L_{m},L_{n}]=(m-n)L_{m+n}$

(so an element of ${\mathcal{W}}$ is an infinite series
$\sum_{n\geq-N}c_{n}L_{n}$).

Let us consider the induced $\operatorname{\mathbb{Z}}(\\!(q)\\!)$-linear
bracket on ${\mathcal{W}}\otimes\operatorname{\mathbb{Z}}(\\!(q)\\!)$. We are
interested in a completion of a certain
$\operatorname{\mathbb{Z}}[\\![q]\\!]$-lattice in
${\mathcal{W}}\otimes\operatorname{\mathbb{Z}}(\\!(q)\\!)$.

######  Definition 2.1.

Let us consider the $\operatorname{\mathbb{Z}}[\\![q]\\!]$-submodule
${\mathcal{W}}^{0}\subset{\mathcal{W}}\otimes\operatorname{\mathbb{Z}}(\\!(q)\\!)$
with the basis $(M_{n})_{n\in\operatorname{\mathbb{Z}}}$, where

$M_{n}=L_{n}\text{ for }n\geq 0,\ \ M_{-n}=q^{n}L_{-n}\text{ for }n>0.$

We also set
${\mathcal{W}}^{0}_{n}:={\mathcal{W}}^{0}/q^{n+1}{\mathcal{W}}^{0}$. Now let
$\widehat{{\mathcal{W}}}^{0}$ (resp., $\widehat{{\mathcal{W}}}^{0}_{n}$)
denote the completion of ${\mathcal{W}}^{0}$ (resp., ${\mathcal{W}}^{0}_{n}$)
with respect to the filtration by
$\operatorname{\mathbb{Z}}[\\![q]\\!]$-submodules generated by $M_{n}$ with
$|n|>N$, where $N=1,2,\ldots$.

In other words, in $\widehat{{\mathcal{W}}}^{0}$ we allow infinite sums in
both directions, $\sum_{i\in\operatorname{\mathbb{Z}}}c_{i}M_{i}$. Using the
relation

$[M_{i},M_{-j}]=q^{\min(i,j)}(i+j)M_{i-j},\text{ for }i\geq 0,j\geq 0,$

it is easy to see that the Lie bracket on ${\mathcal{W}}^{0}$ induces a well
defined Lie bracket on $\widehat{{\mathcal{W}}}^{0}$. We are mostly interested
in the induced Lie algebra structure on $\widehat{{\mathcal{W}}}^{0}_{n}$,
linear over $A_{n}:=\operatorname{\mathbb{Z}}[q]/(q^{n+1})$.

Note that we have a natural involution preserving the bracket,

$\kappa:\widehat{{\mathcal{W}}}^{0}\to\widehat{{\mathcal{W}}}^{0}:\kappa(M_{n})=-M_{-n}\text{
for }n\in\operatorname{\mathbb{Z}}.$ (2.1)

Also, the Lie algebra $\widehat{{\mathcal{W}}}^{0}_{n}$ has a natural
equivariant structure with respect to rescaling of $q$. In other words, if we
think of $\widehat{{\mathcal{W}}}^{0}_{n}$ as a sheaf of Lie algebras over
${\mathbb{A}}^{1}_{q}$ then it has a natural ${\mathbb{G}}_{m}$-equivariant
structure. Namely, for any $A_{n}$-algebra $R$ and any unit $\lambda\in
R^{*}$, we denote by $R(\lambda q)$ the same ring $R$ with the new
$A_{n}$-algebra structure that replaces $q\in R$ by $\lambda q$. Then we have
an action of $\lambda$,

$\widehat{{\mathcal{W}}}^{0}\otimes_{A_{n}}R\rTo{\alpha_{\lambda}}\widehat{{\mathcal{W}}}^{0}\otimes_{A_{n}}R(\lambda
q)$

such that $\alpha_{\lambda}(M_{n})=\lambda^{n}M_{n}$ for $n\geq 0$ and
$\alpha_{\lambda}(M_{n})=M_{n}$ for $n<0$.

Given a $\operatorname{\mathbb{Z}}[q]/(q^{n+1})$-algebra $R$, we are going to
show that the natural Lie action of $\widehat{{\mathcal{W}}}^{0}_{n}$ on
$R(\\!(t)\\!)$ (where $M_{i}$ with $i\leq-n-1$ acts trivially) integrates to
an action of a certain group scheme.

For a $\operatorname{\mathbb{Z}}[q]/(q^{n+1})$-algebra $R$ let us set

$A(R):=R[\\![t_{1},t_{2}]\\!]/(t_{1}t_{2}-q).$

######  Definition 2.2.

We define the group ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ as the group
of continuous automorphisms $\alpha$ of $A(R)$, identical on $R$, such that

$\alpha(t_{1})=t_{1}\cdot u,\ \ \alpha(t_{2})=t_{2}\cdot u^{-1},$

for some unit $u$ in $A(R)$.

The fact that the condition defining
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ is closed under product is easy
to check: if

$\beta(t_{1})=t_{1}\cdot v,\ \ \beta(t_{2})=t_{2}\cdot v^{-1},$

then

$(\alpha\circ\beta)(t_{1})=t_{1}\cdot(u\cdot\alpha(v)),\ \
(\alpha\circ\beta)(t_{2})=t_{2}\cdot(u^{-1}\cdot\alpha(v)^{-1}),$

and we observe that $u\cdot\alpha(v)$ is again a unit in $A(R)$.

We have a natural involution

$\kappa:{\mathcal{G}}_{{\operatorname{glue}},n}(R)\to{\mathcal{G}}_{{\operatorname{glue}},n}(R):\alpha\mapsto\sigma\alpha\sigma^{-1},$

where $\sigma$ swaps $t_{1}$ and $t_{2}$.

In the case $n=0$, i.e., $q=0$, the group
${\mathcal{G}}_{{\operatorname{glue}},0}(R)$ is generated by two commuting
subgroups, $\operatorname{Aut}R[\\![t_{1}]\\!]$ and
$\operatorname{Aut}R[\\![t_{2}]\\!]$ (given by the condition that $u$ depends
only on $t_{1}$ or only on $t_{2}$), which intersect by $R^{*}$. In fact, it
is easy to see that there is an isomorphism of groups

${\mathcal{G}}_{{\operatorname{glue}},0}(R)\simeq\operatorname{Aut}_{1}R[\\![t_{1}]\\!]\times
R^{*}\times\operatorname{Aut}_{1}R[\\![t_{2}]\\!],$

where
$\operatorname{Aut}_{1}R[\\![t]\\!]\subset\operatorname{Aut}R[\\![t]\\!]$
denotes the subgroup of automorphisms $\alpha$ such that $\alpha(t)\equiv
t\mod(t^{2})$.

We have a natural homomorphism of $R$-algebras

$\iota:A(R)\to R(\\!(t_{1})\\!)\oplus
R(\\!(t_{2})\\!):t_{1}\mapsto(t_{1},q/t_{2}),\ t_{2}\mapsto(q/t_{1},t_{2}).$
(2.2)

Note that this is well-defined since $q^{n+1}=0$ in $R$.

Furthermore, it is easy to see that there is a natural identification
$A(R)[t_{i}^{-1}]\simeq R(\\!(t_{i})\\!)$, for $i=1,2$, so that $\iota$ can be
identified with the map

$A(R)\to A(R)[t_{1}^{-1}]\oplus A(R)[t_{2}^{-1}].$

######  Lemma 2.3.

(i) The morphism $\iota$ is injective (in fact, a direct summand as an
$R$-submodule), and we have an inclusion

$R[\\![t_{1}]\\!]t_{1}^{n+1}\oplus
R[\\![t_{2}]\\!]t_{2}^{n+1}\subset\operatorname{im}(\iota).$

(ii) If an element $u\in A(R)^{*}$ satisfies $t_{1}\cdot u=t_{1}$, $t_{2}\cdot
u^{-1}=t_{2}$, then $u=1$. The functor
$R\mapsto{\mathcal{G}}_{{\operatorname{glue}},n}(R)$ is represented by a flat
group scheme ${\mathcal{G}}_{{\operatorname{glue}},n}$ over
$S_{n}=\operatorname{Spec}(\operatorname{\mathbb{Z}}[q]/(q^{n+1}))$ (which as
a scheme is isomorphic to the product of ${\mathbb{G}}_{m}$ with the infinite
dimensional affine space over $S_{n}$). One has

${\mathcal{G}}_{{\operatorname{glue}},n+1}\times
S_{n+1}S_{n}\simeq{\mathcal{G}}_{{\operatorname{glue}},n}.$

###### Proof.

(i) First we observe $\iota(x_{1}^{i})=(x_{1}^{i},0)$,
$\iota(x_{2}^{i})=(0,x_{2}^{i})$ for $i\geq n+1$, which implies the required
inclusion into $\operatorname{im}(\iota)$. Thus, it is enough to prove that
the map

$\overline{\iota}:R[t_{1},t_{2}]/(t_{1}^{n+1},t_{2}^{n+1},t_{1}t_{2}-q)\to
P:=R(\\!(t_{1})\\!)/R[\\![t_{1}]\\!]t_{1}^{n+1}\oplus
R(\\!(t_{2})\\!)/R[\\![t_{2}]\\!]t_{2}^{n+1},$

induced by $\iota$ is injective. Next, let $P^{\prime}$ be the $R$-submodule
of $P$ with the basis $(1,0)$, $(x_{1}^{-i},x_{2}^{-i})_{i\leq 1}$. Then the
$R$-module $P/P^{\prime}$ is free of finite rank, and it is enough to prove
that the map

$R[t_{1},t_{2}]/(t_{1}^{n+1},t_{2}^{n+1},t_{1}t_{2}-q)\to P/P^{\prime}$

of free $R$-modules is an isomorphism. But this is true, since it becomes an
isomorphism modulo $q$, and $q$ is nilpotent.

(ii) Indeed, for such an element we see that $u-1$ goes to zero under the
localization with respect to $t_{1}$ and under the localization with respect
to $t_{2}$. Hence, $\iota(u-1)=0$, so $u=1$. It follows that the elements of
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ are in bijection with $A(R)^{*}$.
Note that an element $u$ of $A(R)$ is a unit if and only if it has form

$u_{0}+a_{1}t_{1}+a_{2}t_{1}^{2}+\ldots+b_{1}t_{2}+b_{1}t_{2}^{2}+\ldots,$

where $u_{0}\in R^{*}$, $a_{i}\in R$, $b_{i}\in R$. This gives an isomorphism
of ${\mathcal{G}}_{{\operatorname{glue}},n}$ with the product of
${\mathbb{G}}_{m}$ and an infinite-dimensional affine space over $S_{n}$. The
last assertion is clear. ∎

It is clear that any $\alpha\in{\mathcal{G}}_{{\operatorname{glue}},n}(R)$
induces well defined homomorphisms $\alpha_{i}$ of $A(R)[t_{i}^{-1}]\simeq
R(\\!(t_{i})\\!)$, so that

$(\alpha_{1},\alpha_{2})\circ\iota=\iota\circ\alpha.$

In this way we get an injective morphism of group ind-schemes

${\mathcal{G}}_{{\operatorname{glue}},n}(R)\to({\mathcal{G}}\times{\mathcal{G}})_{S_{n}}(R)=\operatorname{Aut}R(\\!(t_{1})\\!)\times\operatorname{Aut}R(\\!(t_{2})\\!):\alpha\mapsto(\alpha_{1},\alpha_{2}).$
(2.3)

More explicitly, if $\alpha(t_{1})=t_{1}\cdot u(t_{1},t_{2})$ then

$\alpha_{1}(t_{1})=t_{1}\cdot u(t_{1},q/t_{1}),\ \
\alpha_{2}(t_{2})=t_{2}\cdot u(q/t_{2},t_{2})^{-1}.$

In particular, this map defines an action of
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ on $R(\\!(t_{1})\\!)$ and on
$R(\\!(t_{2})\\!)$. We can think of this geometrically as two actions of
${\mathcal{G}}_{{\operatorname{glue}},n}$ on the punctured formal disk.

######  Example 2.4.

For $r\in R$, we can consider the element
$\alpha\in{\mathcal{G}}_{{\operatorname{glue}},1}(R)$ corresponding to
$u=1+rt_{1}$. Then we have

$\alpha_{1}(t_{1})=t_{1}+rt_{1}^{2},\ \ \alpha_{2}(t_{2})=t_{2}-qr$

(where we use the vanishing of $q^{2}$), so $\alpha_{1}$ acts trivially on the
tangent space of the formal disk, while $\alpha_{2}$ corresponds to the
infinitesimal translation.

For a unit $\lambda\in R^{*}$ let us denote by $R(\lambda q)$ the same ring
$R$ but with the $\operatorname{\mathbb{Z}}[q]/(q^{n+1})$-algebra structure
given by $\lambda\cdot q$. Then we have a natural map

$a_{\lambda}:{\mathcal{G}}_{{\operatorname{glue}},n}(R)\to{\mathcal{G}}_{{\operatorname{glue}},n}(R(\lambda
q))$

sending $u(t_{1},t_{2})\in A(R)^{*}$ to $u(\lambda t_{1},t_{2})\in A(R(\lambda
q))^{*}$. This defines a ${\mathbb{G}}_{m}$-equivariant structure on
${\mathcal{G}}_{{\operatorname{glue}},n}$, so that it descends to a group
scheme over
$\operatorname{Spec}(\operatorname{\mathbb{Z}}[q]/(q^{n+1}))/{\mathbb{G}}_{m}$.

Let us denote by ${\mathcal{G}}_{1}\subset{\mathcal{G}}$ the subgroup given by
${\mathcal{G}}_{1}(R)=\operatorname{Aut}_{1}R[\\![t]\\!]$.

######  Proposition 2.5.

(i) The Lie algebra of ${\mathcal{G}}_{{\operatorname{glue}},n}$ can be
naturally identified with $\widehat{{\mathcal{W}}}^{0}_{n}$, compatibly with
the ${\mathbb{G}}_{m}$-equivariant structures and with their actions on the
punctured disks, so that the Lie algebra involution (2.1) is induced by the
group involution $\kappa$.

(ii) There are natural embedding of groups

$j_{1}:{\mathcal{G}}_{1}(R)=\operatorname{Aut}_{1}R[\\![t_{1}]\\!]\to{\mathcal{G}}_{{\operatorname{glue}},n}(R),\
\
j_{2}:{\mathcal{G}}^{\prime}_{1}(R)=\operatorname{Aut}_{1}R[\\![t_{2}]\\!]\to{\mathcal{G}}_{{\operatorname{glue}},n}(R),\
\ R^{*}\to{\mathcal{G}}_{{\operatorname{glue}},n}(R),$

corresponding to the cases when the unit $u\in A(R)$ depends only on $t_{1}$,
only on $t_{2}$, or is in $R^{*}$, respectively. The product map

$\operatorname{Aut}_{1}R[\\![t_{1}]\\!]\times
R^{*}\times\operatorname{Aut}_{1}R[\\![t_{2}]\\!]\to{\mathcal{G}}_{{\operatorname{glue}},n}(R)$
(2.4)

is an isomorphism of schemes. Consider the subgroups
$\operatorname{Aut}_{n}(R[\\![t]\\!],n)\subset\operatorname{Aut}_{1}(R[\\![t]\\!])$
consisting of automorphisms trivial modulo $t^{n+1}$. Then the map

$\operatorname{Aut}_{n}(R[\\![t_{1}]\\!])\times\operatorname{Aut}_{n}(R[\\![t_{2}]\\!])\to{\mathcal{G}}_{{\operatorname{glue}},n}(R)$

is an embedding of a subgroup, and its composition with the embedding to
${\mathcal{G}}(R)$ is the product of the natural embeddings of the factors.

###### Proof.

(i) We have to consider elements of
${\mathcal{G}}_{{\operatorname{glue}},n}(D)$, where
$D=\operatorname{\mathbb{Z}}[q,\epsilon]/(q^{n+1},\epsilon^{2})$, reducing to
the identity modulo $\epsilon$, which are given by units of the form
$u=1+\epsilon a$, where $a\in A(\operatorname{\mathbb{Z}}[q]/(q^{n+1}))$. The
basis element $M_{i}$ for $i\geq 0$ correspond to $u_{i}=1+\epsilon
t_{1}^{i}$, while $M_{-i}$ for $i\geq 0$ corresponds to $u_{-i}=1+\epsilon
t_{2}^{i}$. A direct computation of the commutators shows that these elements
satisfy the relations of $\widehat{{\mathcal{W}}}^{0}_{n}$.

(ii) To prove that (2.4) is injective we need to check that the intersection
of subgroups $\operatorname{Aut}_{1}R[\\![t_{1}]\\!]$ and
$\operatorname{Aut}R[\\![t_{2}]\\!]$ in
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ is trivial. Indeed, the
corresponding unit $u\in A(R)^{*}$ belongs to at the same time to
$1+t_{1}R[\\![t_{1}]\\!]$ and to $R[\\![t_{2}]\\!]$, hence $u=1$.

To check surjectivity, let us use the induction on $n$. If $q=0$ then the
assertion is clear. Assume that $n\geq 1$ and we know the result modulo
$q^{n}$, so for any element $g\in{\mathcal{G}}_{{\operatorname{glue}},n}(R)$,
there exists a decomposition $g\mod q^{n}=g_{1}\cdot g_{2}$, where $g_{1}$ and
$g_{2}$ belong to the $R/(q^{n})$-points of the corresponding subgroups.
Lifting $g_{1}$ and $g_{2}$ to $R$-points $\widetilde{g}_{1}$ and
$\widetilde{g}_{2}$ and replacing $g$ by
$\widetilde{g}_{1}^{-1}g\widetilde{g}_{2}^{-1}$, we reduce to the case when
$g\mod q^{n}=1$. Thus, we need to show that every unit $u\in A(R)^{*}$ of the
form $u=1+q^{n}a$, can be represented in the form

$u=u_{1}\cdot\alpha_{u_{1}}(u_{2}),$

where $u_{1}\in 1+t_{1}R[\\![t_{1}]\\!]$, $u_{2}\in R[\\![t_{2}]\\!]$ and
$\alpha_{u_{1}}$ is an automorphism of $A(R)$ associated with $u_{1}$. We can
write $a=a_{1}+a_{2}$, where $a_{1}\in t_{1}R[\\![t_{1}]\\!]$ and $a_{2}\in
R[\\![t_{2}]\\!]$. Now we can take $u_{1}=1+q^{n}a_{1}$, $u_{2}=1+q^{n}a_{2}$.
Note that $q^{2n}=0$, so $u_{1}^{-1}=1-q^{n}a_{1}$ and since
$q^{n}t_{1}t_{2}=0$, we deduce that $\alpha_{u_{1}}$ acts trivially on
$R[\\![t_{2}]\\!]$. Thus,

$u_{1}\cdot\alpha_{u_{1}}(u_{2})=u_{1}\cdot u_{2}=1+q^{n}(a_{1}+a_{2})=u.$

The last assertion is straightforward. ∎

###  2.2. Isomorphisms between node deformations

######  Proposition 2.6.

Suppose $q,q^{\prime}\in R$ are two nilpotent elements. Then any continuous
isomorphism of $R$-algebras,

$\alpha:R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q^{\prime})\to
R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)$

sending $(x_{1})$ to $(x_{1})$ and $(x_{2})$ to $(x_{2})$, is a composition of
an isomorphism

$x_{1}\mapsto\lambda x_{1},\ x_{2}\mapsto x_{2},$

for uniquely defined $\lambda\in R^{*}$, where $q^{\prime}=\lambda q$,
followed by an automorphism from ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$.

###### Proof.

Since $\alpha((x_{1}x_{2}))=(x_{1}x_{2})$, we have the equality of ideals
$(q)=(q^{\prime})$, so $\alpha$ induces an automorphism of
$R/q[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2})$.

By assumption, we have

$\alpha(x_{1})=x_{1}u_{1}(x_{1},x_{2}),\ \
\alpha(x_{2})=x_{2}u_{2}(x_{1},x_{2}).$

Reducing modulo $q$, we deduce that $u_{1}=\lambda_{1}\mod(x_{1},x_{2})$,
$u_{2}=\lambda_{2}\mod(x_{1},x_{2})$, where $\lambda_{1}$ and $\lambda_{2}$
project to units in $R/(q)$. Hence, $\lambda_{1}$ and $\lambda_{2}$ are units
in $R$, and so $u_{1}$ and $u_{2}$ are units in $A(R)$. Post-composing
$\alpha$ with an element of ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$
corresponding to $u_{2}$, we obtain an isomorphism

$\alpha^{\prime}(x_{1})=x_{1}u(x_{1},x_{2}),\ \ \alpha^{\prime}(x_{2})=x_{2},$

where $u=u_{1}u_{2}$ is a unit in $R$. Let us write

$u=\lambda+x_{1}f_{1}(x_{1})+x_{2}f_{2}(x_{2}),$

where $\lambda\in^{*}$. Then the condition

$q^{\prime}=\alpha^{\prime}(x_{1})\alpha^{\prime}(x_{2})=x_{1}x_{2}u=qu$

implies that

$q^{\prime}=\lambda q,\ \ qf_{1}=0,\ \ qf_{2}=0.$

It follows that

$x_{1}x_{2}f_{2}=0,$

so we can replace $u$ with $u^{\prime}=\lambda+x_{1}f_{1}(x_{1})$ and still
have $\alpha^{\prime}(x_{1})=x_{1}u^{\prime}$. Since we also have

$x_{2}x_{1}f_{1}=0,$

if we write $u^{\prime}=\lambda\cdot u_{1}$, then we get

$x_{2}\cdot(u_{1}-1)=0.$

Hence,

$x_{2}\cdot u_{1}^{-1}=x_{2}+x_{2}(u_{1}^{-1}-1)=x_{2},$

and the automorphism $x_{1}\mapsto x_{1}\cdot u_{1}$, $x_{2}\mapsto x_{2}$
belongs to ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$. Thus,
$\alpha^{\prime}$ is the composition of the isomorphism $x_{1}\mapsto\lambda
x_{1}$, $x_{2}\mapsto x_{2}$ with an element of
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$, as claimed.

To prove the uniqueness of $\lambda$ we have to check that if

$\lambda\cdot x_{1}=x_{1}\cdot u,\ \ x_{2}=x_{2}\cdot u^{-1}$

for some unit $u$ in $A(R)$ and some $\lambda\in R^{*}$, then $\lambda=1$.
Writing

$u=\lambda_{1}+x_{1}f_{1}(x_{1})+x_{2}f_{2}(x_{2}),$

where $\lambda_{1}\in R^{*}$, we get that

$\lambda\cdot x_{1}=\lambda_{1}+x_{1}^{2}f_{1}(x_{1})+qf_{2}(x_{2}),$

which implies that $f_{1}=0$. Hence $u^{-1}$ has form

$u^{-1}=\lambda_{1}^{-1}+x_{2}\widetilde{f}_{2}(x_{2}).$

But then the condition $x_{2}=x_{2}\cdot u^{-1}$ implies that
$\widetilde{f}_{2}=0$ and $\lambda_{1}=1$. Hence $u^{-1}=1$ and $\lambda=1$. ∎

##  3\. Extended clutching construction

###  3.1. Construction

Let $\pi:\widetilde{C}_{0}\to\operatorname{Spec}(R)$ be a proper flat family
of curves over an affine base, equipped with a pair of distinct sections
$p_{1},p_{2}:\operatorname{Spec}(R)\to\widetilde{C}_{0}$, such that $\pi$ is
smooth along the images of $p_{1}$ and $p_{2}$ (abusing the notation we will
denote these images also by $p_{1}$ and $p_{2}$). Then one can define a new
family $C_{0}\to\operatorname{Spec}(R)$ obtained from $\widetilde{C}_{0}$ by
gluing $p_{1}$ with $p_{2}$ into a node.

We are interested in a generalization of this construction that gives a
deformation of the nodal curve $C_{0}$. For this we fix an additional data,
namely, the formal parameters $x_{1}$ and $x_{2}$ at $p_{1}$ and $p_{2}$,
respectively.

We also assume that an element $q\in R$ is fixed such that $q^{n+1}=0$. Let us
denote by $C_{0,R/q}\to\operatorname{Spec}(R/q)$ the curve obtained from
$C_{0}$ by the base change $R\to R/q$. We will construct a family of curves
${\mathcal{C}}$ over $R$, deforming $C_{0,R/q}$. Loosely speaking,
${\mathcal{C}}$ will be glued from
$\widetilde{C}_{0}\setminus\\{p_{1},p_{2}\\}$ and
$\operatorname{Spf}(R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q))$ along
$\operatorname{Spf}(R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!))$, i.e., the union
of formal punctured disks in $\widetilde{C}_{0}$ around $p_{1}$ and $p_{2}$.

Let us give a more precise construction of ${\mathcal{C}}$. We use the
homomorphism of $R$-algebras,

$\iota:A(R)=R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)\to R(\\!(x_{1})\\!)\oplus
R(\\!(x_{2})\\!):f(x_{1},x_{2})\mapsto(f(x_{1},q/x_{1}),f(q/x_{2},x_{2}))$

(see (2.2)). Recall that by Lemma 2.3, $\iota$ is injective and

$R[\\![x_{1}]\\!]x_{1}^{n+1}\oplus
R[\\![x_{2}]\\!]x_{2}^{n+1}\subset\operatorname{im}(\iota).$ (3.1)

Let $U\subset\widetilde{C}_{0}$ be an affine neighborhood of
$\\{p_{1},p_{2}\\}$ in $\widetilde{C}_{0}$. The expansion in formal parameters
gives a homomorphism

$\kappa:{\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})\to R(\\!(x_{1})\\!)\oplus
R(\\!(x_{2})\\!).$

######  Lemma 3.1.

The map

${\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})\to(R(\\!(x_{1})\\!)\oplus
R(\\!(x_{2})\\!))/\operatorname{im}(\iota).$

induced by $\kappa$, is surjective.

###### Proof.

First, we note that the image of $\iota$ is contained in
$x_{1}^{-n}R[\\![x_{1}]\\!]\oplus x_{2}^{-n}R[\\![x_{2}]\\!]$. Thus, it is
enough to check that for each $N\geq n$, the map

$H^{0}(U,{\mathcal{O}}_{U}(Np_{1}+Np_{2}))\to x_{1}^{-N}R[\\![x_{1}]\\!]\oplus
x_{2}^{-N}R[\\![x_{2}]\\!]/\operatorname{im}(\iota)$

is surjective. Note that due to the inclusion (3.1), the target of the above
map is finitely generated as $R$-module. Thus, it is enough to check
surjectivity modulo $q$. But then the required surjectivity follows from
surjectivity of the map on global sections induced by the morphism of coherent
sheaves on $U$,

${\mathcal{O}}_{U}(Np_{1}+Np_{2})\to{\mathcal{O}}_{U}(Np_{1}+Np_{2})/{\mathcal{O}}_{U}(-p_{1}-p_{2}).$

∎

Now let us define the $R$-subalgebra $A$ in
${\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})$ as the following fibered product:

$\begin{diagram}$ (3.2)

######  Lemma 3.2.

Let $f\in{\mathcal{O}}(U)$ be such that
$\kappa(f)=(x_{1}^{n+1}+\ldots,x_{2}^{n+1}+\ldots)$. Then $f\in A$ and the
localization $A[f^{-1}]$ is naturally isomorphic to ${\mathcal{O}}(U\setminus
Z(f))$, where $Z(f)$ is the divisor of zeros of $f$. Locally near
$\\{p_{1},p_{2}\\}$, the divisor $Z(f)$ is supported on $\\{p_{1},p_{2}\\}$.

###### Proof.

For the first assertion, it is enough to check that the map obtained from
$\iota$ by localization with respect to $f$ is an isomorphism. Since the
multiplication by $\iota(f)$ is invertible on $R(\\!(x_{1})\\!)\oplus
R(\\!(x_{2})\\!)$, we have to show that for every element $(p,q)\in
R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!)$, one has
$\iota(f^{N})(p,q)\in\operatorname{im}(\iota)$ for some $N$. But this
immediately follows from the inclusion (3.1).

For the last assertion, we can assume that $q=0$. It is enough to check that
locally near $\\{p_{1},p_{2}\\}$, one has an inclusion of ideals
${\mathcal{O}}_{U}(-(n+1)p_{1}-(n+1)p_{2})\subset(f)$. But this follows from
the fact that this inclusion holds in a formal neighborhood of
$\\{p_{1},p_{2}\\}$. ∎

Lemma 3.2 implies that one can shrink $U$ so that $Z(f)$ is supported at
$\\{p_{1},p_{2}\\}$ and the morphism

$U\setminus\\{p_{1},p_{2}\\}\to\operatorname{Spec}(A)$

is an open embedding. Therefore, we can define the $R$-scheme ${\mathcal{C}}$
by gluing two open subsets, $\widetilde{C}_{0}\setminus\\{p_{1},p_{2}\\}$ and
$\operatorname{Spec}(A)$ along $U\setminus\\{p_{1},p_{2}\\}$.

######  Proposition 3.3.

The scheme
${\mathcal{C}}={\mathcal{C}}(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2};q)$ is
flat and proper over $\operatorname{Spec}(R)$. It does not depend on a choice
of an affine neighborhood $U$ up to a canonical isomorphism. Given a
homomorphism $\phi:R\to R^{\prime}$, setting $q^{\prime}=\phi(q)$, we have a
natural isomorphism

${\mathcal{C}}(\widetilde{C}_{0}\times_{\operatorname{Spec}R}\operatorname{Spec}R^{\prime},p^{\prime}_{1},p^{\prime}_{2};q^{\prime})\simeq{\mathcal{C}}(\widetilde{C}_{0},p_{1},p_{2};q)\times_{\operatorname{Spec}R}\operatorname{Spec}R^{\prime},$

where $p^{\prime}_{1}$ and $p^{\prime}_{2}$ are obtained from $p_{1}$ and
$p_{2}$ by the base change. In particular, the base change
${\mathcal{C}}\times_{\operatorname{Spec}R}\operatorname{Spec}R/q$ gives the
curve $C_{0,R/q}$ obtained by gluing the marked points $p_{1}$ and $p_{2}$ in
$\widetilde{C}_{0,R/q}$ into a node $\nu\in C_{0,R/q}$.

###### Proof.

To prove flatness it is enough to check that $A$ is flat over $R$. But this
follows from the exact sequence of $R$-modules

$0\to
A\to{\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})\to\operatorname{coker}(\iota)\to
0,$

where surjectivity follows from Lemma 3.1. Since $\operatorname{coker}(\iota)$
is a free $R$-module, it follows that $A$ is flat over $R$.

Next, to show properness, it is enough to consider the corresponding family
over $\operatorname{Spec}(R/q)$, i.e., we can assume that $q=0$. In this case
the image of $\iota$ is contained in $R[\\![x_{1}]\\!]\oplus
R[\\![x_{2}]\\!]\subset R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!)$, so $A$ is
contained in
${\mathcal{O}}(U)\subset{\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})$. This
implies that we have an affine morphism $f:\widetilde{C}_{0}\to{\mathcal{C}}$,
which is an isomorphism over
$\widetilde{C}_{0}\setminus\\{p_{1},p_{2}\\}\subset{\mathcal{C}}$. It is
enough to check that in fact $f$ is a finite morphism. Indeed, we need to
check that ${\mathcal{O}}(U)$ is a finite $A$-module. From the definition of
$\iota$, we see that $A$ is exactly the subring of $\phi\in{\mathcal{O}}(U)$
such that $p_{1}^{*}\phi=p_{2}^{*}\phi\in R$. Thus, choosing a function
$\phi_{0}\in{\mathcal{O}}(U)$ such that $p_{1}^{*}\phi_{0}=1$ and
$p_{2}^{*}\phi_{0}=0$ (such $\phi_{0}$ exists since $U$ is affine), we obtain

${\mathcal{O}}(U)=A\oplus R\cdot\phi_{0},$

which implies that ${\mathcal{O}}(U)$ is a finite $A$-module, as claimed.

Let us see what happens if we replace $U$ by a smaller neighborhood
$U^{\prime}\subset U$ of $\\{p_{1},p_{2}\\}$. Let us denote the corresponding
glued curve by ${\mathcal{C}}(U^{\prime})$. Note that we have a natural
morphism ${\mathcal{C}}(U^{\prime})\to{\mathcal{C}}={\mathcal{C}}(U)$.

We can choose a function $g$ on $U$ such that $g|_{U\setminus U^{\prime}}=0$
and

$\kappa(g)=(1+O(x_{1}^{n+1}),1+O(x_{2}^{n+1})).$

This implies that $g$ belongs $A$ and the localization $A_{g}$ fits into a
fibered product similar to (3.2), but with $U$ replaced by $U\setminus
Z(g)\subset U^{\prime}$, where $Z(g)$ is the zero locus of $g$. Hence, the
morphism ${\mathcal{C}}(U\setminus Z(g))\to{\mathcal{C}}$ is an open
embedding. Since it is also proper, it is an embedding of a connected
component. Looking at the induced map over $R/q$, we see that it is an
isomorphism. Thus, the map ${\mathcal{C}}(U^{\prime})\to{\mathcal{C}}$ has the
right inverse ${\mathcal{C}}\simeq{\mathcal{C}}(U\setminus
Z(g))\to{\mathcal{C}}(U^{\prime})$. Repeating the same argument with $U$
replaced by $U^{\prime}$ and $U^{\prime}$ replaced by $U\setminus Z(g)$, we
see that ${\mathcal{C}}(U^{\prime})\to{\mathcal{C}}$ is an isomorphism.

Finally, to show the compatibility with the base change $R\to R^{\prime}$, it
is enough to check the corresponding base change of the diagram (3.2) is still
cartesian. But this follows from the fact that the bottom arrow of this
diagram can be replaced by a map of free $R$-modules, upon taking quotients by
$(x_{1}^{n+1},x_{2}^{n+1})$. ∎

Note that we have a natural closed embedding $\nu:\operatorname{Spec}(R/q)\to
C_{0,R/q}\subset{\mathcal{C}}$, which corresponds to the composed homomorphism

$A\rTo{\widetilde{\kappa}}R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)\to R/q$

sending $x_{1}$ and $x_{2}$ to $0$. The curve ${\mathcal{C}}$ comes with a
flat covering $U_{1}\sqcup U_{2}\to{\mathcal{C}}$, where
$U_{1}=\widetilde{C}_{0}\setminus\\{p_{1},p_{2}\\}$, and
$U_{2}=\operatorname{Spf}(R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q))$, where we
identify $U_{2}$ with the formal neighborhood of of the node $\nu$.

From the above construction we obtain morphisms

$\overline{{\mathcal{M}}}_{g_{1},1}^{(\infty)}\times\overline{{\mathcal{M}}}^{(\infty)}_{g_{2},1}\times\operatorname{Spec}\operatorname{\mathbb{Z}}[q]/(q^{n+1})\to\overline{{\mathcal{M}}}_{g},$
(3.3)
$\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}\times\operatorname{Spec}\operatorname{\mathbb{Z}}[q]/(q^{n+1})\to\overline{{\mathcal{M}}}_{g},$
(3.4)

for $g_{1}\geq 1$, $g_{2}\geq 1$, $g_{1}+g_{2}=g\geq 2$. Namely, for a ring
$R$, an $R$-object of
$\overline{{\mathcal{M}}}_{g_{1},1}^{(\infty)}\times\overline{{\mathcal{M}}}^{(\infty)}_{g_{2},1}\times\operatorname{Spec}\operatorname{\mathbb{Z}}[q]/(q^{n+1})$
gives a pair of curves with marked points $(C_{1},p_{1})$, $(C_{2},p_{2})$
over $R$ and an element $q_{R}\in R$ such that $q_{R}^{n+1}=0$. Thus, we can
apply the above construction to the curve $\widetilde{C}_{0}=C_{1}\sqcup
C_{2}$, the marked points $p_{1},p_{2}$, and the element $q_{R}$. The
corresponding glued curve ${\mathcal{C}}$ over $R$ will be stable of genus
$g$, so we obtain an object of $\overline{{\mathcal{M}}}_{g}(R)$. The
definition of the second morphism is similar.

###  3.2. Recovering the gluing data

######  Proposition 3.4.

The flat proper family of curves,
${\mathcal{C}}={\mathcal{C}}(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2};q)$,
constructed in Proposition 3.3, is equipped with a closed embedding
$\nu:\operatorname{Spec}(R/(q))\hookrightarrow{\mathcal{C}}$ of $R$-schemes
and an isomorphism of $R$-algebras

$\eta:R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)\rTo{\sim}{\mathcal{O}}(\hat{{\mathcal{C}}}_{\nu}),$

compatible with homomorphisms to $R/(q)$, where $\hat{{\mathcal{C}}}_{\nu}$ is
the completion of ${\mathcal{C}}$ along $\nu(\operatorname{Spec}(R/q))$. We
further require that the ideal of the image corresponds under $\eta$ to the
ideal $(x_{1},x_{2})$. The correspondence

$(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2})\mapsto({\mathcal{C}},\nu,\eta)$

is an equivalence of categories fibered over the category of
$\operatorname{\mathbb{Z}}/(q^{n+1})$-algebras.

###### Proof.

Step 1. For the first statement we have to check that the map
$\widetilde{\kappa}$ in the fibered square (3.2) induces an isomorphism of the
completion of $A$ at the ideal $I=\widetilde{\kappa}^{-1}((x_{1},x_{2}))$. Let
us choose an element $f_{N}\in{\mathcal{O}}(U\setminus\\{p_{1},p_{2}\\})$ such
that $\kappa(f_{N})=(x_{1}^{N}+\ldots,x_{2}^{N}+\ldots)$ for very large $N$.
Then $f_{N}\in A$ and for any $a\in A$ we have

$\kappa(a\cdot f_{N})\subset\operatorname{im}(\iota)\cdot\kappa(f_{N})\subset
R[\\![x_{1}]\\!]x_{1}^{n+1}\oplus
R[\\![x_{2}]\\!]x_{2}^{n+1}\subset\iota((x_{1},x_{2})),$

hence, $(f_{N})\subset I$. On the other hand, we have

$\kappa\iota((x_{1},x_{2})^{2N})\subset(R[\\![x_{1}]\\!]x_{1}^{n+1}\oplus
R[\\![x_{2}]\\!]x_{2}^{n+1})\cdot\kappa(f_{N})\subset\kappa(A\cdot f_{N}).$

Therefore, $I^{2N}\subset(f_{N})$, and so the ideals $(f_{N})$ and $I$ define
equivalent topologies on $A$. Now if we pass to completions with respect to
the $(f_{N})$-adic topology in the cartesian square (3.2), the the map
$\kappa$ will become an isomorphism. Hence, $\widetilde{\kappa}$ will also be
an isomorphism after completion, as claimed.

Step 2. Now let us start with a flat proper family of curves ${\mathcal{C}}$
over $\operatorname{Spec}(R)$, together with the data $(\nu,\eta)$ as in the
statement. Let $V=\operatorname{Spec}(A)$ be an open affine neighborhood of
the image of $\nu$, and let $I\subset A$ be the ideal of
$\nu(\operatorname{Spec}(R/q))$, so that $\eta$ gives an isomorphism

$R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)\simeq\hat{A}_{I}=\varprojlim_{n}A/I^{n}.$

In particular, for each $N>0$ we have a surjective map

$A\to A/I^{N}\simeq(R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q))/(x_{1},x_{2})^{N}.$

We can pick $f\in A$ such that $f$ maps to $x_{1}^{n+1}+x_{2}^{n+1}$ under
this map for $N>n+1$. Then we have an induced map

$A[f^{-1}]\to R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)[f^{-1}]\simeq
R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!)$

(see the proof of Lemma 3.2).

Now let us define the algebra $B$ from the cartesian square where the bottom
horizontal arrow is the natural embedding. To see that $B$ is flat over $R$,
it is enough to check that the map

$A[f^{-1}]\to(R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!))/(R[\\![x_{1}]\\!]\oplus
R[\\![x_{2}]\\!])$

is surjective. For this we can assume that $q=0$. For every $k>0$ there exists
an element $g\in A$ such that $g$ mapsto
$x_{1}^{d(n+1)-k}\mod(x_{1},x_{2})^{d(n+1)}$ in
$R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2},x_{1}^{d(n+1)},x_{2}^{d(n+1)})$. It
follows that $gf^{-d(n+1)}$ maps to an element in
$x_{1}^{-k}+x_{1}^{-k+1}R[\\![x_{1}]\\!]+R[\\![x_{2}]\\!]$ in
$R(\\!(x_{1})\\!)\oplus R(\\!(x_{2})\\!)$. Similarly, there exists an element
of $A[f^{-1}]$ mapping to an element in
$x_{2}^{-k}+R[\\![x_{1}]\\!]+x_{2}^{-k+1}R[\\![x_{2}]\\!]$, which proves the
claimed surjectivity.

Step 3. Let $J\subset B$ denote the kernel of the homomorphism $B\to
R[\\![x_{1}]\\!]\oplus R[\\![x_{2}]\\!]\to R\oplus R$. We claim that the
completion of $B$ with respect to the $J$-adic topology is precisely
$R[\\![x_{1}]\\!]\oplus R[\\![x_{2}]\\!]$. Indeed, this is proved in the same
way as in Step 1, by choosing an element $f_{N}$ mapping to
$x_{1}^{N}+x_{2}^{N}$ for a very large $N$.

Next, we observe that an element $f\in A\subset A[f^{-1}]$ belongs to $B$ and
it is easy to see that $B[f^{-1}]\simeq A[f^{-1}]$. Hence, we can view
$\operatorname{Spec}(A[f^{-1}])=V\setminus Z(f)$ as an open subscheme of
$\operatorname{Spec}(B)$.

We claim that shrinking $V$ we can assume that $Z(f)$ is supported at
$\operatorname{im}(\nu)$. Indeed, it is enough to check that $(f)$ contains
$I^{N}$ for some $N$, in a formal neighborhood of $\operatorname{im}(\nu)$.
But this follows from the fact that the ideal generated by
$(x_{1}^{n+1}+\ldots,x_{2}^{n+1}+\ldots)$ in
$R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)$ contains $(x_{1},x_{2})^{N}$ for
sufficiently large $N$.

Now, shrinking $V$ as above, we define the curve $\widetilde{C}_{0}$ by gluing
the $R$-schemes $\operatorname{Spec}(B)$ and
${\mathcal{C}}\setminus\operatorname{im}(\eta)$ along the
$V\setminus\operatorname{im}(\eta)=\operatorname{Spec}(B[f^{-1}])$. By
construction we have a pair of disjoint sections
$p_{1},p_{2}:\operatorname{Spec}(R)\to\operatorname{Spec}(B)$, such that the
completion of $B$ along each gives an algebra isomorphic to
$R[\\![x_{i}]\\!]$. As in the proof of Prop. 3.3, one can show that
$\widetilde{C}_{0}$ is proper over $\operatorname{Spec}(R)$ and that this
construction does not depend on a choice of the neighborhood $V$.

It is easy to see that the two constructions are inverse of each other, so
this establishes the claimed equivalence. ∎

###  3.3. Computing the determinant line bundle

Recall that the boundary line bundle ${\mathcal{O}}(\Delta)$ on
$\overline{{\mathcal{M}}}_{g}$ is defined as the determinant line bundle

${\mathcal{L}}_{C/S}:=\det\bigl{(}R\pi_{*}(\Omega_{C/S}\to\omega_{C/S})[1]\bigr{)}$

associated with any family of stable curves $\pi:C\to S$. This line bundle is
equipped with a canonical section $\theta$ (see [1, Sec. XIII.4] which is a
special case of a general construction of [4]). We will compute the pair
$({\mathcal{L}}_{C/S},\theta)$ for the family corresponding to the extended
clutching construction.

######  Proposition 3.5.

For the family
${\mathcal{C}}={\mathcal{C}}(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2};q)$
over $\operatorname{Spec}(R)$, there is a natural trivialization

$\tau:R\rTo{\sim}{\mathcal{L}}_{{\mathcal{C}}/\operatorname{Spec}(R)}$

such that $\theta=\tau(q)$.

###### Proof.

We can compute the object
$R\pi_{*}[\Omega_{{\mathcal{C}}/S}\to\omega_{{\mathcal{C}}/S}]$ using Cech
resolutions

$\Omega_{{\mathcal{C}}/S}\to\Omega_{U_{1}/S}\oplus\Omega_{U_{2}/S}\to\Omega_{U_{12}/S},$
$\omega_{{\mathcal{C}}/S}\to\omega_{U_{1}/S}\oplus\omega_{U_{2}/S}\to\omega_{U_{12}/S}.$

Since the maps $\Omega_{U_{1}/S}\to\omega_{U_{1}/S}$ and
$\Omega_{U_{12}/S}\to\omega_{U_{12}/S}$ are isomorphisms, we get a natural
quasi-isomorphism

$R\pi_{*}(\Omega_{C/S}\to\omega_{C/S})\to[\Omega_{U_{2}/S}\to\omega_{U_{2}/S}].$

Hence,

${\mathcal{L}}_{{\mathcal{C}}/S}\simeq\det\bigl{(}[\Omega_{U_{2}/S}\to\omega_{U_{2}/S}][1]\bigr{)}$

and $\theta$ is the determinant of the map $A\to B$, where $[A\to B]$ is a
complex of projective finitely generated $R$-modules quasi-isomorphic to
$[\Omega_{U_{2}/S}\to\omega_{U_{2}/S}]$.

Now using the coordinates on $U_{2}=\operatorname{Spf}({\mathcal{A}})$, where
${\mathcal{A}}=R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)$, we get

$\Omega_{U_{2}/S}={\mathcal{A}}\cdot dx_{1}\oplus{\mathcal{A}}\cdot
dx_{2}/(x_{1}dx_{2}+x_{2}dx_{1}),$ $\omega_{U_{2}/S}={\mathcal{A}}\cdot e,$

where $e=-\frac{dx_{1}}{x_{1}}=\frac{dx_{2}}{x_{2}}$. It is easy to see that
we have decompositions of $R$-modules

$\Omega_{U_{2}/S}=R\cdot x_{1}dx_{2}\oplus R[\\![x_{1}]\\!]\cdot dx_{1}\oplus
R[\\![x_{2}]\\!]\cdot dx_{2},$ $\omega_{U_{2}/S}=R\cdot e\oplus
x_{1}R[\\![x_{1}]\\!]\cdot e\oplus x_{2}R[\\![x_{2}]\\!]\cdot e,$

and the complex $[\Omega_{U_{2}/S}\to\omega_{U_{2}/S}]$ splits into a direct
sum of

$[R\cdot x_{1}dx_{2}\to R\cdot e]\simeq[R\rTo{q}R]$

and of an acyclic complex. Hence, the element

$e\otimes(x_{1}dx_{2})^{-1}\in\det[R\cdot x_{1}dx_{2}\to R\cdot e]$

indfuces the required trivialization $\tau$ of
${\mathcal{L}}_{{\mathcal{C}}/S}$ such that $\tau(q)=\theta$. ∎

##  4\. The formal neighborhood of the boundary divisor and the gluing group
scheme

###  4.1. Changes of formal parameters on curves

Let $(C,p)$ be a family of curves over a commutative ring $R$, with a marked
point $p\in C(R)$, such that $C$ is smooth over $R$ near $p$.

Let us denote by $\hat{{\mathcal{O}}}_{C,p}$ the ring of functions on the
completion of $C$ near $p$:

$\hat{{\mathcal{O}}}_{C,p}:=\varprojlim_{n}H^{0}(C,{\mathcal{O}}_{C}/{\mathcal{O}}_{C}(-np)).$

Recall that a relative formal parameter $t$ at $p$ is an element of ideal of
$p$ in $\hat{{\mathcal{O}}}_{C,p}$ inducing an isomorphism

$R[\\![t]\\!]\rTo{\sim}\hat{{\mathcal{O}}}_{C,p}.$

Then we have the induced isomorphism

$R(\\!(t)\\!)\rTo{\sim}{\mathcal{K}}_{C,p},$

where
${\mathcal{K}}_{C,p}:=\varinjlim_{m}\varprojlim_{n}H^{0}(C,{\mathcal{O}}_{C}(mp)/{\mathcal{O}}_{C}(-np)$.

Given $(C,p,t)$, where $t$ is a formal parameter at $p$, and a continuous
automorphism $\alpha$ of $R(\\!(t)\\!)$, we can define new data
$(C^{\prime},p^{\prime},t^{\prime})$ as follows. Set $U$ be an open affine
neighborhood of $p$. Then $C$ is glued from $U$ and $C\setminus p$ along
$U\setminus p$. We have natural homomorphisms

${\mathcal{O}}(U)\to\hat{{\mathcal{O}}}_{C,p},\ \ {\mathcal{O}}(U\setminus
p)\to{\mathcal{K}}_{C,p},$

so that we have a cartesian diagram Using the formal parameter $t$ we can
replace the bottom row in this diagam by the map
$R[\\![t]\\!]\to{\mathcal{K}}_{C,p}$.

Now we define $C^{\prime}$ by gluing the new affine curve $U^{\prime}$ with
$C\setminus p$ along $U\setminus p$, where ${\mathcal{O}}(U^{\prime})$ is
defined as the fibered product

$\begin{diagram}$ (4.1)

The composed homomorphism ${\mathcal{O}}(U^{\prime})\to R[\\![t]\\!]\to R$ of
$R$-algebras defines a marked point $p^{\prime}$ on $U^{\prime}\subset
C^{\prime}$, such that $U^{\prime}\setminus p^{\prime}\simeq U\setminus p$
(and $C^{\prime}\setminus p^{\prime}\simeq C\setminus p$).

It is easy to see that the construction of $C^{\prime}$ does not depend on a
choice of $U$, and defines an action of the ind-group scheme ${\mathcal{G}}$
on the moduli stack $\overline{{\mathcal{M}}}^{(\infty)}_{g,1}$ of curves $C$
with a smooth marked point $p$ and a formal parameter $t$ at $p$ (see also [2,
Sec. 17.3]). Similarly, ${\mathcal{G}}\times{\mathcal{G}}$ acts on
$\overline{{\mathcal{M}}}^{(\infty)}_{g,2}$ by regluing at two marked marked
points.

Thus, using homomorphism (2.3) from ${\mathcal{G}}_{{\operatorname{glue}},n}$
to $({\mathcal{G}}\times{\mathcal{G}})_{S_{n}}$, we get an action of
${\mathcal{G}}_{{\operatorname{glue}},n}$ on

$\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2};n}=\overline{{\mathcal{M}}}_{g_{1},1}^{(\infty)}\times\overline{{\mathcal{M}}}_{g_{2},1}^{(\infty)}\times
S_{n}.$

###  4.2. The use of translations in genus $1$

In this section we work characteristic zero, i.e., we assume the base ring $R$
to be a $\operatorname{\mathbb{Q}}$-algebra. For genus $1$ we have the
following notion of canonical formal parameter.

######  Definition 4.1.

Let $(C,p)$ be a stable curve of genus $1$ with one marked point over
$S=\operatorname{Spec}(R)$. We say that a formal parameter $t$ at $p$ is
canonical if the formal differential $dt$ extends to a global section of the
dualizing sheaf $\omega_{C/S}$.

Since for $(C,p)$ stable of genus $1$ the map
$H^{0}(C,\omega_{C})\to\omega_{C}|_{p}$ is an isomorphism, for every formal
parameter $t$ at $p$ there exists a unique canonical formal parameter $t_{c}$
at $p$ such that $t_{c}\equiv t\mod(t^{2})$.

Let us consider the subgroup
${\mathcal{G}}^{\prime}_{1}\subset{\mathcal{G}}_{{\operatorname{glue}},1}$
acting on ${\mathcal{M}}^{(\infty)}_{1,h;1}$ by the change the formal
parameters

$(x_{1},x_{2})\mapsto(x_{1}-qc_{2},x_{2}+c_{2}x_{2}^{2}+\ldots)$ (4.2)

and changing the marked point $p_{1}$ accordingly (see Prop. 2.5(ii)). We
claim that if we assume that $x_{1}$ is canonical and $q^{2}=0$, then this
action is equivalent to the standard action of ${\mathcal{G}}^{\prime}_{1}$
via the second factor.

######  Lemma 4.2.

Let $(C,p)$ be a stable pointed curve of genus $1$ over $R$, $t$ a canonical
formal parameter at $p$, and $r\in R$ an element such that $r^{2}=0$. Let us
consider a reglued data $(C,p^{\prime},t^{\prime})$, where $t^{\prime}=t+r$.
Then there is a natural isomorphism $(C,p,t)\simeq(C,p^{\prime},t^{\prime})$.

###### Proof.

First, we observe that since $\omega_{C/S}$ is isomorphic to the pull-back of
a line bundle on $S$, the natural map of $R$-modules

$H^{0}(C,\omega^{-1}_{C/S})\to\omega^{-1}_{C/S}|_{p}$

is an isomorphism. Hence, there exists an element $\widetilde{v}\in
H^{0}(C,\omega^{-1}_{C/S})$ whose pairing with $dt\in H^{0}(C,\omega_{C/S})$
is equal to $1$. Let $v\in H^{0}(C,{\mathcal{T}}_{C/S})$ denote the global
derivation of ${\mathcal{O}}_{C}$, which is the image of $\widetilde{v}$ under
the map $\omega^{-1}_{C/S}\to{\mathcal{T}}_{C/S}$, dual to the natural map
$\Omega_{C/S}\to\omega_{C/S}$ (which is an isomorphism away from the nodes).
Then we still have $v(t)=1$.

Next, we observe that since $r^{2}=0$, we have an automorphism
$\exp(rv):f\mapsto f+rv(f)$ of $C$, acting trivially on the underlying
topological space. Thus, if $U$ is an open affine neighborhood of $p$, then we
have a commutative square

Thus, if ${\mathcal{O}}(U^{\prime})$ is defined by the cartesian square (4.1),
with $\alpha(t)=t+r$, then $\exp(rv)$ induces an isomorphism between
${\mathcal{O}}(U)$ and ${\mathcal{O}}(U^{\prime})$, compatible with the marked
points $p$ and $p^{\prime}$, and fitting into a commutative square Together
with the automorphism $\exp(rv)$ of $C\setminus p$, this gives an isomorphism
$C\rTo{\sim}C^{\prime}$ sending $p$ to $p^{\prime}$ and $t$ to
$t^{\prime}=t+r$. ∎

The proof of Theorem 1.2 will be based on Propositions 4.3 and 4.5 below.

######  Proposition 4.3.

For any $h\geq 1$, there is a natural isomorphism

$\overline{{\mathcal{M}}}^{(\infty)}_{1,h;1}/{\mathcal{G}}_{{\operatorname{glue}},1}\simeq(\overline{{\mathcal{M}}}^{(1)}_{1,1}\times\overline{{\mathcal{M}}}_{h,1}^{(1)}\times
S_{1})/{\mathbb{G}}_{m},$

where ${\mathbb{G}}_{m}$ changes $(t_{1},t_{2})$ to $(\lambda
t_{1},\lambda^{-1}t_{2})$.

######  Lemma 4.4.

Let a group scheme $G$ act on a scheme $X$. Assume that $G$ has two subgroups
$H_{1},H_{2}\subset G$, such that the product map

$H_{1}\times H_{2}\rTo{\sim}G$

is an isomorphism of schemes. Assume also that $Y\subset X$ is an
$H_{2}$-invariant subscheme, such that the $H_{1}$-action induces an
isomorphism of schemes

$H_{1}\times Y\rTo{\sim}X,$

i.e., $Y$ is a section for the (free) action of $H_{1}$ on $X$. Then we have a
natural isomorphism of quotient stacks

$[Y/H_{2}]{\rTo{\sim}}[X/G].$

###### Proof.

It suffices to prove that the map induced by the $G$-action,

$G\times_{H_{2}}Y\to X,$

is an isomorphism. We claim that the composition

$X\to H_{1}\times Y\to G\times Y\to G\times_{H_{2}}Y$

is the inverse. Indeed, starting from $(g,y)\in G\times Y$, consider the
unique decomposition of $gy$,

$gy=h_{1}\cdot y^{\prime},$

with $h_{1}\in H_{1}$ and $y^{\prime}\in Y$. On the other hand, we have a
unique decomposition

$g=h^{\prime}_{1}\cdot h_{2},$

where $h^{\prime}_{1}\in H_{1}$, $h_{2}\in H$. Now, since $h_{2}y\in Y$, the
equality

$h^{\prime}_{1}\cdot(h_{2}y)=h_{1}\cdot y^{\prime}$

implies that $h^{\prime}_{1}=h_{1}$ and $y^{\prime}=h_{2}y$. Hence,
$(g,y)=(h_{1}h_{2},h_{2}^{-1}y^{\prime})$ projects to the same point as
$(h_{1},y^{\prime})$ in $G\times_{H_{2}}Y$. ∎

###### Proof of Proposition 4.3.

Let us apply Lemma 4.4 to $X={\mathcal{M}}^{(\infty)}_{1,h;1}$,
$G={\mathcal{G}}_{{\operatorname{glue}},1}$, $Y\subset X$ the subscheme
corresponding to the data with the parameter $x_{1}$ canonical,
$H_{2}={\mathbb{G}}_{m}\times{\mathcal{G}}^{\prime}_{1}$, where
${\mathcal{G}}^{\prime}_{1}$ acts by (4.2), and $H_{1}={\mathcal{G}}_{1}$ the
subgroup corresponding to reversing the roles of $x_{1}$ and $x_{2}$. Note
that ${\mathcal{G}}_{1}$ acts freely on ${\mathcal{M}}^{(\infty)}_{1,1}$ and
the subscheme with $x_{1}$ canonical is a section of this action. This implies
that $H_{1}$ acts freely on $X$ and $Y$ is a section for this action. On the
other hand, by Lemma 4.2, the action of $H_{2}$ preserves $Y$, and we get an
isomorphism

${\mathcal{M}}^{(\infty)}_{1,h;1}/{\mathcal{G}}_{{\operatorname{glue}},1}\simeq(\overline{{\mathcal{M}}}^{(1)}_{1,1}\times\overline{{\mathcal{M}}}_{g_{2},1}^{(\infty)}\times
S_{1})/({\mathbb{G}}_{m}\times{\mathcal{G}}^{\prime}_{1}),$

where ${\mathcal{G}}^{\prime}_{1}$ acts only on the factor
$\overline{{\mathcal{M}}}_{g_{2},1}^{(\infty)}$. Thus, taking the quotient by
the action of ${\mathcal{G}}^{\prime}_{1}$ has the effect of replacing
$\overline{{\mathcal{M}}}_{g_{2},1}^{(\infty)}$ with
$\overline{{\mathcal{M}}}_{g_{2},1}^{(1)}$. ∎

######  Proposition 4.5.

For any $n\geq 0$, there are natural isomorphisms

$\overline{{\mathcal{M}}}^{(\infty)}_{1,1;n}/{\mathcal{G}}_{{\operatorname{glue}},n}\simeq(\overline{{\mathcal{M}}}^{(1)}_{1,1}\times\overline{{\mathcal{M}}}_{1,1}^{(1)}\times
S_{n})/{\mathbb{G}}_{m},$ $(\overline{{\mathcal{M}}}^{(\infty)}_{1,2}\times
S_{n})/{\mathcal{G}}_{{\operatorname{glue}},n}\simeq(\overline{{\mathcal{M}}}^{(1)}_{1,2}\times
S_{n})/{\mathbb{G}}_{m},$

where ${\mathbb{G}}_{m}$ changes $(t_{1},t_{2})$ to $(\lambda
t_{1},\lambda^{-1}t_{2})$.

###### Proof.

We will give the proof of the first isomorphism. The proof of the second
isomorphism is similar.

We identify
$Y_{n}:=\overline{{\mathcal{M}}}^{(1)}_{1,1}\times\overline{{\mathcal{M}}}_{1,1}^{(1)}\times
S_{n}$ with a closed subscheme in $X_{n}:={\mathcal{M}}^{(\infty)}_{1,1;n}$
defined by the condition that both parameters $t_{1}$ and $t_{2}$ are
canonical. Note that the embedding $Y_{n}\subset X_{n}$ is a section of the
natural projection $X_{n}\to Y_{n}$. It suffices to prove that the morphism
given by the action,

${\mathcal{G}}_{{\operatorname{glue}},n}\times_{{\mathbb{G}}_{m}}Y_{n}\to
X_{n},$

is an isomorphism.

We will prove this by induction on $n$. The case $n=0$ is clear, since in this
case the subgroup
${\mathcal{G}}_{1}\times{\mathcal{G}}_{1}\subset{\mathcal{G}}_{{\operatorname{glue}},0}$
acts separately on two factors ${\mathcal{M}}^{(\infty)}_{1,1}$. Now assume
$n>0$ and the assertion holds for $n-1$. We use the fact that
${\mathcal{G}}_{{\operatorname{glue}},n}$ is a smooth group scheme over
$S_{n}$, and that
$(X_{n-1},Y_{n-1},{\mathcal{G}}_{{\operatorname{glue}},n-1})$ are obtained by
the base change $S_{n-1}\to S_{n}$ from
$(X_{n},Y_{n},{\mathcal{G}}_{{\operatorname{glue}},n})$ in a way compatible
with all the structures.

Given an object $t_{\bullet}=(C_{1},p_{1},t_{1};C_{2},p_{2},t_{2};q)$ of
$X_{n}(R)$ we can consider the reduction modulo $q^{n}$,

$\overline{t}_{\bullet}\in X_{n}(R/(q^{n}))=X_{n-1}(R/(q^{n})).$

By assumption, there exists an element
$\overline{g}\in{\mathcal{G}}_{{\operatorname{glue}},n-1}(R/(q^{n}))={\mathcal{G}}_{{\operatorname{glue}},n}(R/(q^{n}))$
such that $\overline{g}\cdot\overline{t}_{\bullet}$ is in
$Y_{n-1}(R/(q^{n}))$. Let us lift $\overline{g}$ to an element
$g\in{\mathcal{G}}_{{\operatorname{glue}},n}(R)$, and set
$t^{\prime}_{\bullet}=g\cdot t_{\bullet}\in X_{n}(R)$. Then the reductions of
$t^{\prime}_{1}$ and $t^{\prime}_{2}$ modulo $q^{n}$ are canonical parameters.
Let $t_{1,c}$ and $t_{2,c}$ be the canonical parameters for
$(C^{\prime}_{1},p^{\prime}_{1})$ and $C^{\prime}_{2},p^{\prime}_{2}$ such
that $t_{i,c}$ agrees with $t^{\prime}_{i}$ modulo the square of the maximal
ideal of $p^{\prime}_{i}$. Then we obtain that

$t_{1,c}=t^{\prime}_{1}+q^{n}r_{1}(t^{\prime}_{1})^{2}+\ldots,\ \
t_{2,c}=t^{\prime}_{2}+q^{n}r_{2}(t^{\prime}_{2})^{2}+\ldots,$

where all the higher coefficients are divisible by $q^{n}$. But this gives an
element

$g^{\prime}\in\operatorname{Aut}_{1}(R(\\!(t_{1})\\!),n)\times\operatorname{Aut}_{1}(R(\\!(t_{2})\\!),n)\subset{\mathcal{G}}_{{\operatorname{glue}},n}(R)$

such that $g^{\prime}(t^{\prime}_{1},t^{\prime}_{2})=(t_{1,c},t_{2,c})$ (here
we use the last part of Proposition 2.5(ii)).

On the other hand, given an $R$-point
$g\in{\mathcal{G}}_{{\operatorname{glue}},n}(R)$ and an object
$t_{\bullet}=(C_{1},p_{1},t_{1};C_{2},p_{2},t_{2};q)$ of $Y_{n}(R)$, such that
$g\cdot t_{\bullet}$ is in $Y_{n}(R)$, we need to check that $g$ is in
${\mathbb{G}}_{m}(R)\subset{\mathcal{G}}_{{\operatorname{glue}},n}(R)$. By the
induction assumption, this is true modulo $q^{n}$, so changing $g$ by an
element of ${\mathbb{G}}_{m}(R)$, we can assume that $g\equiv 1\mod q^{n}$. We
can write $g$ in the form $g=g_{1}\cdot g_{2}\cdot\lambda$, with $g_{i}\in
j_{i}(\operatorname{Aut}_{1}R(\\!(t_{i})\\!))$, $\lambda\in R^{*}$. Then we
have $g_{i}(t_{i})\equiv t_{i}\mod q^{n}$ and $\lambda\equiv 1\mod q^{n}$.
Note that $\lambda\cdot t_{\bullet}$ is still in $Y_{n}(R)$, while the action
of $g_{1}\cdot g_{2}$ changes the parameters $(t_{1},t_{2})$ to
$(g_{1}(t_{1}),g_{2}(t_{2}))$. Since these should be canonical, we deduce that
$g_{1}=g_{2}=1$, as claimed. ∎

###  4.3. Connection with a ${\mathbb{G}}_{m}$-torsor over
${\mathcal{E}}_{g_{1},g_{2}}^{(n)}$

Let $R$ be a commutative $\operatorname{\mathbb{Z}}[q]/(q^{n+1})$-algebra,
$(\widetilde{C}_{0},p_{1},p_{2})$ a family of curves over $R$ with two
distinct smooth marked points and with relative formal parameters $x_{i}$ at
$p_{i}$. The extended clutching construction from Sec. 3 associates with these
data a curve
${\mathcal{C}}={\mathcal{C}}(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2};q)$
over $R$, which is equipped with a closed embedding
$\nu:\operatorname{Spec}(R/q)\hookrightarrow{\mathcal{C}}$ and an isomorphism
of $R$-algebras

$\eta:R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)\rTo{\sim}{\mathcal{O}}(\hat{{\mathcal{C}}}_{\nu}).$

Now we observe that the group ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$
acts on the gluing data through the homomorphism
${\mathcal{G}}_{{\operatorname{glue}},n}(R)\to\operatorname{Aut}R(\\!(t_{1})\\!)\times\operatorname{Aut}R(\\!(t_{1})\\!)$
and the action of the latter group on the data
$(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2})$ (see Sec. 4.1). On the other
hand, the group ${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ clearly acts on
the set of possible isomorphisms $\eta$ (for a fixed curve ${\mathcal{C}}$ and
fixed $\nu$). We claim that these two actions are compatible.

######  Lemma 4.6.

For any $\operatorname{\mathbb{Z}}[q]/(q^{n+1})$-algebra $R$, the equivalence
between the data $(\widetilde{C}_{0},p_{1},p_{2},x_{1},x_{2})$ and
$({\mathcal{C}},\nu,\eta)$ (see Prop. 3.4) is compatible with the
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$-action. Furthermore, the
trivialization
$\tau=\tau_{x_{1},x_{2},q}:R\to{\mathcal{L}}_{{\mathcal{C}}/\operatorname{Spec}(R)}$
defined in Proposition 3.5, is preserved by the
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$-action and satisfies

$\tau_{\lambda x_{1},x_{2},\lambda q}=\tau_{x_{1},\lambda x_{2},\lambda
q}=\lambda^{-1}\cdot\tau_{x_{1},x_{2},q}$

for $\lambda\in R^{*}$.

###### Proof.

Recall that the curve ${\mathcal{C}}$ is obtained by gluing
$\widetilde{C}_{0}\setminus\\{p_{1},p_{2}\\}$ with $\operatorname{Spec}(A)$,
where the $R$-algebra $A$ is defined as the fibered product (3.2), and the
isomorphism $\eta$ is induced by the homomorphism $\widetilde{\kappa}:A\to
R[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)$. We can combine the cartesian square
(3.2) defining $A$ with a commutative square for any continuous automorphism
$\alpha$ of $R[\\![x_{1},x_{2}]\\!]$ preserving the ideals $(x_{1})$,
$(x_{2})$ and inducing the automorphisms $\alpha_{i}$ of $R(\\!(x_{i})\\!)$.
In the case when $\alpha\in{\mathcal{G}}_{{\operatorname{glue}},n}(R)$, this
immediately shows the claimed compatibility of the equivalence of Prop. 3.4
with the ${\mathcal{G}}_{{\operatorname{glue}},n}$-action.

Next, we want to check that ${\mathcal{G}}_{{\operatorname{glue}},n}$-action
preserves the trivialization of the determinant bundle given in Proposition
3.5. For an automorphism $\alpha\in{\mathcal{G}}_{{\operatorname{glue}},n}$
such that $\alpha(x_{1})=u^{-1}\cdot x_{1}$ and $\alpha(x_{2})=u\cdot x_{2}$,
for some unit $u$ in $R[\\![x_{1},x_{2}]\\!]$, one can easily check that

$\alpha(\frac{dx_{2}}{x_{2}})=U\cdot\frac{dx_{2}}{x_{2}},\ \
\alpha(x_{1}dx_{2})=U\cdot x_{1}dx_{2},\ \text{ where}$
$U=1+u^{-1}\frac{\partial u}{\partial x_{2}}x_{2}-u^{-1}\frac{\partial
u}{\partial x_{1}}x_{1}.$

Hence, $\alpha$ preserves the trivialization of determinant of the subcomplex
$[R\cdot x_{1}dx_{2}\to R\cdot e]$ of $[\Omega_{U_{2}/S}\to\omega_{U_{2}/S}]$,
and hence it preserves the trivialization of
${\mathcal{L}}_{{\mathcal{C}}/S}$.

On the other hand, the rescaling $(x_{1},x_{2},q)\mapsto(\lambda\cdot
x_{1},x_{2},\lambda\cdot q)$ does not change the generator $e=dx_{2}/x_{2}$ of
$\omega_{U_{2}/S}$, and rescales $x_{1}dx_{2}$ to $\lambda\cdot x_{1}dx_{2}$.
Hence, the corresponding trivialization of the determinant line bundle gets
rescaled by $\lambda^{-1}$. ∎

Proof of Theorem 1.1. Lemma 4.6 implies that morphisms (3.3), (3.4) factor
through morphisms

$[\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2},n}/{\mathcal{G}}_{{\operatorname{glue}},n}]\to\overline{{\mathcal{M}}}_{g},$
(4.3) $[(\overline{{\mathcal{M}}}^{(\infty)}_{g-1,2}\times
S_{n})/{\mathcal{G}}_{{\operatorname{glue}},n}]\to\overline{{\mathcal{M}}}_{g}$
(4.4)

We claim that the morphism (4.3) (resp., (4.4)) factors through
$\Delta^{(n)}_{g_{1},g_{2}}\subset\overline{{\mathcal{M}}}_{g}$ (resp.,
$\Delta^{(n)}_{0}\subset\overline{{\mathcal{M}}}_{g}$). Indeed, this
immediately follows from Proposition 3.5. Let us consider for example the
boundary divisor $\Delta_{g_{1},g_{2}}$. It is defined as the vanishing locus
of the section $\theta$ of the determinant line bundle, and the pull-back of
this section is $q$. Since $q^{n+1}=0$, we see that (4.3) factors through
$\Delta^{(n)}_{g_{1},g_{2}}\subset\overline{{\mathcal{M}}}_{g}$. Hence, the
defining equation of $\Delta^{(n)}_{g_{1},g_{2}}$ vanishes on the open dense
locus in $\overline{{\mathcal{M}}}^{(\infty)}_{g_{1},g_{2},n}$ corresponding
to pairs of smooth curves, and so it vanishes everywhere.

Next, in the case $g_{1}\neq g_{2}$, the equivalence of Proposition 3.4
induces an equivalence of ${\mathcal{M}}^{(\infty)}_{g_{1},g_{2},n}$ with the
moduli functor ${\mathcal{F}}$, associating to a commutative ring $R$, an
element $q\in R$ such that $q^{n+1}=0$ and the data $({\mathcal{C}},\nu,\eta)$
as in Proposition 3.4, such that ${\mathcal{C}}$ is a family of stable curves
with the unique separating node given by $\nu$ and under $\eta$ the variable
$x_{1}$ corresponds to the branch of genus $g_{1}$. Note that the
${\mathcal{F}}(R)$ is equipped with a natural free action of
${\mathcal{G}}_{{\operatorname{glue}},n}(R)$ by changing the isomorphism
$\eta$. In addition, we have an action of ${\mathbb{G}}_{m}$ on
${\mathcal{F}}$ that changes $(x_{1},x_{2},q)$ to $(\lambda
x_{1},x_{2},\lambda q)$. By Proposition 2.6, the quotient
${\mathcal{M}}^{(\infty)}_{g_{1},g_{2},n}/{\mathcal{G}}_{{\operatorname{glue}},n}$
is a ${\mathbb{G}}_{m}$-torsor over $U\Delta^{(n)}_{g_{1},g_{2}}$ Furthermore,
by Lemma 4.6, we have a morphism of ${\mathbb{G}}_{m}$-torsors from this
quotient to the ${\mathbb{G}}_{m}$-torsor of trivializations of
${\mathcal{L}}_{{\mathcal{C}}/\operatorname{Spec}(R)}$, which is necessarily
an isomorphism. This finishes the proof in the case $g_{1}\neq g_{2}$.

In the case $g_{1}=g_{2}=g/2$, similarly, from Propositions 3.4 and 2.6 we get
that
${\mathcal{M}}^{(\infty)}_{g/2,g/2,n}/{\mathcal{G}}_{{\operatorname{glue}},n}$
(resp., $({\mathcal{M}}^{(\infty)}_{g-1,2}\times
S_{n})/{\mathcal{G}}_{{\operatorname{glue}},n}$) is a
${\mathbb{G}}_{m}$-torsor over ${\mathcal{E}}^{(n)}_{g/2,g/2}$ (resp.,
${\mathcal{E}}^{(n)}_{0}$), and we finish by using Lemma 4.6 as above. ∎

Proof of Theorem 1.2. Let $X_{1,g-1}^{(n)}$ denote the
${\mathbb{G}}_{m}$-torsor over $U\Delta_{1,g-1}^{(n)}$ corresponding to the
line bundle ${\mathcal{O}}(\Delta)$. Combining Theorem 1.1 with Proposition
4.3, we get an isomorphism for $g>2$,

$X_{1,g-1}^{(1)}\simeq{\mathcal{M}}^{(\infty)}_{1,g-1;1}/{\mathcal{G}}_{{\operatorname{glue}},1}\simeq({\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{g-1,1}^{(1)}\times
S_{1})/{\mathbb{G}}_{m}.$

Furthermore, as in the proof of Theorem 1.1, it is compatible with the
${\mathbb{G}}_{m}$-action, where ${\mathbb{G}}_{m}$ acts on
${\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{g-1,1}^{(1)}\times S_{1}$ by
rescaling $(x_{1},x_{2},q)\mapsto(\lambda\cdot x_{1},x_{2},\lambda\cdot q)$.
Let us denote the latter action as $l(G_{m})$, to distinguish it from the
subgroup ${\mathbb{G}}_{m}\subset{\mathcal{G}}_{{\operatorname{glue}},n}$
acting by $(x_{1},x_{2},q)\mapsto(\lambda\cdot x_{1},\lambda^{-1}\cdot
x_{2})$. It follows that

$[X_{1,g-1}^{(1)}/{\mathbb{G}}_{m}]\simeq[[{\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{1,1}^{(1)}\times
S_{n})/{\mathbb{G}}_{m}]/l({\mathbb{G}}_{m})]\simeq[({\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{1,1}^{(1)}\times
S_{1})/(l({\mathbb{G}}_{m})\times r({\mathbb{G}}_{m}))],$ (4.5)

where $r({\mathbb{G}}_{m})$ acts by $(x_{1},x_{2},q)\mapsto(x_{1},\lambda\cdot
x_{2},\lambda\cdot q)$.

Similarly, in the case $g=2$, we have ${\mathbb{G}}_{m}$-torsors
$X_{1,1}^{(n)}\to{\mathcal{E}}_{1,1}^{(n)}$ and
$X_{0}^{(n)}\to{\mathcal{E}}_{0}^{(n)}$, associated with
${\mathcal{O}}(\Delta)$, and we have isomorphisms

$X_{1,1}^{(n)}\simeq{\mathcal{M}}^{(\infty)}_{1,1;1}/{\mathcal{G}}_{{\operatorname{glue}},n}\simeq[({\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{1,1}^{(1)}\times
S_{n})/{\mathbb{G}}_{m}],$
$X_{0}^{(n)}\simeq({\mathcal{M}}^{(\infty)}_{1,2}\times
S_{n})/{\mathcal{G}}_{{\operatorname{glue}},n}\simeq({\mathcal{M}}^{(1)}_{1,2}\times
S_{n})/{\mathbb{G}}_{m},$

compatible with the extra $l({\mathbb{G}}_{m})$-action. As before, this
induces isomorphisms of quotients

$[X_{1,1}^{(n)}/{\mathbb{G}}_{m}]\simeq[({\mathcal{M}}^{(1)}_{1,1}\times{\mathcal{M}}_{1,1}^{(1)}\times
S_{n})/(l({\mathbb{G}}_{m})\times r({\mathbb{G}}_{m}))],$ (4.6)
$[X_{0}^{(n)}/{\mathbb{G}}_{m}]\simeq[({\mathcal{M}}^{(1)}_{1,2}\times
S_{n})/(l({\mathbb{G}}_{m})\times r({\mathbb{G}}_{m}))].$ (4.7)

Now we use the following easy observation: if $L$ is a line bundle over $S$
and ${\mathcal{P}}\to S$ is the corresponding ${\mathbb{G}}_{m}$-torsor, then
the total space of $L$ can be identified with the quotient
$({\mathcal{P}}\times{\mathbb{A}}^{1})/{\mathbb{G}}_{m}$, where
$\lambda\in{\mathbb{G}}_{m}$ acts by $(p,q)\mapsto(\lambda p,\lambda q)$, and
the $n$th infinitesimal neighborhood of the zero section in the total space of
$L$ can be identified with the subscheme

$({\mathcal{P}}\times
S_{n})/{\mathbb{G}}_{m}\subset({\mathcal{P}}\times{\mathbb{A}}^{1})/{\mathbb{G}}_{m}.$

Similarly, if $L_{1},L_{2}$ are two line bundles, ${\mathcal{P}}_{1}$,
${\mathcal{P}}_{2}$ are the corresponding ${\mathbb{G}}_{m}$-torsors, then

$({\mathcal{P}}_{1}\times_{S}{\mathcal{P}}_{2}\times
S_{n})/{\mathbb{G}}_{m}^{2},$

where $(\lambda_{1},\lambda_{2})\in{\mathbb{G}}_{m}^{2}$ acts by
$(p_{1},p_{2},q)\mapsto(\lambda_{1}p_{1},\lambda_{2}p_{2},\lambda_{1}\lambda_{2}q)$,
can be identified with $n$th infinitesimal neighborhood of the zero section in
the total space of $L_{1}\otimes L_{2}$.

To deduce the result, we apply this observation to the right-hand sides of
(4.5), (4.6) and (4.6), and use the well known identification of the normal
bundle with the product of line bundles associated with the marked points (see
[1, Sec. XIII.3]). ∎

##  5\. Periods near a separating node

###  5.1. Differentials and cohomology

In this section we work in the complex analytic category.

Let us consider a family of stable curves $\pi:C\to S$, where $S$ is a disk,
the total space $C$ is smooth, and $\pi$ is smooth over $S\setminus 0$. Let
$\omega_{C/S}$ denote the relative dualizing sheaf. It is well known that in
this case the higher direct images of the relative logarithmic de Rham
complex, which can be identified with

$\overline{{\mathcal{H}}}^{i}:=R^{i}\pi_{*}[{\mathcal{O}}_{C}\to\omega_{C/S}],$

are vector bundles on $S$, which give a canonical extension of the local
systems

${\mathcal{H}}^{i}:=R^{i}\pi_{*}\pi^{-1}{\mathcal{O}}_{S\setminus 0}$

from $S\setminus 0$ (see [7, Thm. 2.18]).

Thus, the coboundary map

$\pi_{*}(\omega_{C/S})\to R^{1}\pi_{*}[{\mathcal{O}}_{C}\to\omega_{C/S}]$

extends the standard embedding of $\pi_{*}\omega_{C/S}$ into
$R^{1}\pi_{*}\pi^{-1}{\mathcal{O}}_{S}$ defining periods over $S\setminus 0$.

Furthermore, we claim that the formation of the higher direct image
$\overline{{\mathcal{H}}}^{1}=R^{1}\pi_{*}[{\mathcal{O}}_{C}\to\omega_{C/S}]$
is compatible with the base change from $S$ to the $n$-th infinitesimal
neighborhoods $S_{n}\subset S$ of $0\in S$. Indeed, even though the
differential ${\mathcal{O}}_{C}\to\omega_{C/S}$ is not
${\mathcal{O}}_{C}$-linear, computing the direct image using a Cech
resolution, one arrives in a standard way to the setup of the cohomology base
change formalism as in [6, Thm. II.5, p.46]. Thus, the computation of the
cohomology of restriction of $[{\mathcal{O}}_{C}\to\omega_{C/S}]$ over the
special fiber (see [7, Thm. 2.18]), implies that we have a cohomology base
change isomorphism

$\overline{{\mathcal{H}}}^{1}|_{S_{n}}\simeq
H^{1}(C_{n},[{\mathcal{O}}_{C_{n}}\to\omega_{C_{n}/S_{n}}]),$

where $C_{n}\to S_{n}$ is the base change family over $S_{n}$. It is also well
known that the formation of $\pi_{*}(\omega_{C/S})$ for families of stable
curves is compatible with the base change.

Now we start with a stable curve $C_{0}$ which is the union of the smooth
components $C_{1}$ and $C_{2}$, such that points $p_{1}\in C_{1}$ and
$p_{2}\in C_{2}$ are glued into a node $p_{0}\in C_{0}$. We consider any
deformation $\pi:{\mathcal{C}}\to S_{n}$ of a stable curve $C_{0}$ over
$S_{n}$, inducing a nontrivial deformation of the node singularity over
$S_{1}$. Such a deformation always arises in a fashion described above, so the
computation of the map

$\Pi:\pi_{*}(\omega_{{\mathcal{C}}/S_{n}})\to
R^{1}\pi_{*}[{\mathcal{O}}_{{\mathcal{C}}}\to\omega_{{\mathcal{C}}/S_{n}}]$
(5.1)

would give us information about the period map near $C_{0}$.

We can compute
$R^{i}\pi_{*}[{\mathcal{O}}_{{\mathcal{C}}}\to\omega_{{\mathcal{C}}/S_{n}}]$
using the Cech complex with respect to an acyclic flat covering
${\mathcal{C}}=U_{1}\cup U_{2}$, where
$U_{1}={\mathcal{C}}\setminus\\{p_{0}\\}$ and $U_{2}$ is the formal
neighborhood of $p_{0}$ in ${\mathcal{C}}$. Note that since
$C_{0}\setminus\\{p_{0}\\}=(C_{1}\setminus\\{p_{1}\\})\sqcup(C_{2}\setminus\\{p_{2}\\})$
is smooth and affine, $U_{1}$ is a trivial deformation of
$C_{0}\setminus\\{p_{0}\\}$. On the other hand, we can identify $U_{2}$ with
the standard deformation
${\mathcal{O}}_{S_{n}}[\\![x_{1},x_{2}]\\!]/(x_{1}x_{2}-q)$, where $q$ is a
coordinate on $S$ (where $x_{i}$ is a formal parameter at $p_{i}\in C_{i}$,
for $i=1,2$). Then
${\mathcal{O}}(U_{12})={\mathcal{O}}_{S_{n}}(\\!(x_{1})\\!)\oplus{\mathcal{O}}_{S_{n}}(\\!(x_{2})\\!)$.

######  Lemma 5.1.

The natural projection induces a quasi-isomorphism

${\mathcal{C}}[{\mathcal{O}}_{\mathcal{C}}\to\omega_{{\mathcal{C}}/S_{n}}]\to[{\mathcal{O}}(U_{1})\rTo{d}\omega_{{\mathcal{C}}/S_{n}}(U_{1})\to\omega_{{\mathcal{C}}/S_{n}}(U_{12})/(d{\mathcal{O}}(U_{12})+\omega_{{\mathcal{C}}/S_{n}}(U_{2}))].$

Furthermore,

$[{\mathcal{O}}(U_{1})\to\omega_{{\mathcal{C}}/S_{n}}(U_{1})]$

is a subcomplex in the above complex, and the embedding induces an isomorphism
on $H^{1}$. Thus, we get an identification

$R^{1}\pi_{*}[{\mathcal{O}}_{\mathcal{C}}\to\omega_{{\mathcal{C}}/S_{n}}]\simeq\omega_{{\mathcal{C}}/S_{n}}(U_{1})/d{\mathcal{O}}(U_{1}).$

###### Proof.

First, let us consider the short exact sequence of complexes, where
${\mathcal{C}}^{\bullet}={\mathcal{C}}[{\mathcal{O}}_{\mathcal{C}}\to\omega_{{\mathcal{C}}/S_{n}}]$.
Since the first row is exact, the first assertion follows from the fact that
the map

$d:{\mathcal{O}}(U_{12})/{\mathcal{O}}(U_{2})\to\omega_{{\mathcal{C}}/S_{n}}(U_{12})/\omega_{{\mathcal{C}}/S_{n}}(U_{2})$

is injective. The latter injectivity can be checked as follows: both modules
are free over ${\mathcal{O}}(S_{n})=\operatorname{\mathbb{C}}[t]/(t^{n})$, so
the injectivity follows from the injectivity modullo $(t)$.

Recall that $U_{1}$ is a trivial deformation of
$(C_{1}\setminus\\{p_{1}\\})\sqcup(C_{2}\setminus\\{p_{2}\\})$. The map
$\omega_{{\mathcal{C}}/S_{n}}(U_{1})\to\omega_{{\mathcal{C}}/S_{n}}(U_{12})$
sends a pair of diffferentials $(\omega_{1},\omega_{2})$, where $\omega_{i}\in
H^{0}(C_{i}\setminus\\{i\\},\omega_{C_{i}})\otimes\operatorname{\mathbb{C}}[t]/(t^{n})$,
for $i=1,2$, to their expansions in the formal parameters at $p_{1}$, $p_{2}$.
By the residue theorem, the coefficients of $dx_{1}/x_{1}$ and $dx_{2}/x_{2}$
in these expansions will be trivial. Hence, the map

$\omega_{{\mathcal{C}}/S_{n}}(U_{1})\to\omega_{{\mathcal{C}}/S_{n}}(U_{12})/d{\mathcal{O}}(U_{12})$

is zero, which implies the second assertion. ∎

Using Lemma 5.1 and the triviality of the deformation $U_{1}/S_{n}$ we can
view the map (5.1) as a morphism of
$\operatorname{\mathbb{C}}[q]/(q^{n+1})$-modules

$\begin{array}[]{l}\Pi:H^{0}({\mathcal{C}},\omega_{{\mathcal{C}}/S_{n}})\to\omega_{{\mathcal{C}}/S_{n}}(U_{1})/d{\mathcal{O}}(U_{1})\simeq\\\
\bigl{(}\omega_{C_{1}}(C_{1}\setminus\\{p_{1}\\})/d{\mathcal{O}}(C_{1}\setminus\\{p_{1}\\})\oplus\omega_{C_{2}}(C_{2}\setminus\\{p_{2}\\})/d{\mathcal{O}}(C_{2}\setminus\\{p_{2}\\})\bigr{)}\otimes\operatorname{\mathbb{C}}[q]/(q^{n+1}),\end{array}$
(5.2)

induced by the restriction map
$H^{0}({\mathcal{C}},\omega_{{\mathcal{C}}/S_{n}})\to\omega_{{\mathcal{C}}/S_{n}}(U_{1})/d{\mathcal{O}}(U_{1})$.
Note since the map $\Pi$ in injective modulo $q$, it defines a split embedding
of free $\operatorname{\mathbb{C}}[q]/(q^{n+1})$-modules.

###  5.2. Basis of differentials regular outside a point

Let $C$ be a smooth projective curve of genus $g\geq 1$, $p\in C$ a point such
that $h^{1}({\mathcal{O}}(gp))=0$ (i.e., $p$ is not a Weierstrass point), and
$z$ a formal parameter at $p$.

######  Lemma 5.2.

(i) There is a unique basis $\alpha[0],\ldots,\alpha[g-1]$ of regular
differentials on $C$ with $\alpha[i]=(z^{i}+O(z^{g}))dz$, for
$i=0,\ldots,g-1$.

(ii) For each $n\geq 2$, there is a unique rational differential
$\omega[-n]\in H^{0}(C,\omega_{C}(np))$ with $\omega[-n]=(z^{-n}+O(z^{g}))dz$.
The differentials
$(\alpha[0],\ldots,\alpha[g-1],\omega[-2],\omega[-3],\ldots)$ form a basis of
$\omega_{C}(C\setminus p)$.

(iii) For every nonzero tangent vector $v$ at $p$ there is a unique formal
parameter $z$ at $p$ such that $v(t)=1$ and $z^{g-1}dz$ extends to a regular
differential on $C$, i.e., $\alpha[g-1]=z^{g-1}dz$.

###### Proof.

(i) The condition $H^{1}({\mathcal{O}}(gp))=0$ implies that
$H^{0}(\omega_{C}(-gp))=0$ and $h^{1}(\omega_{C}(-gp))=1$. Hence, for every
$i=0,\ldots,g$, we have $h^{0}(\omega_{C}(-(g-i)p))=i$. Now we can prove by
induction on $i$ that there is a unique basis $\alpha[g-i],\ldots,\alpha[g-1]$
of $H^{0}(\omega_{C}(-(g-i)p))$ with $\alpha[j]=(z^{j}+O(z^{g}))dz$. For $i=0$
such a basis is empty. Assume we have such a basis of
$H^{0}(\omega_{C}(-(g-i)p))$. Let us choose an element $\alpha\in
H^{0}(\omega_{C}(-(g-i+1)p))\setminus H^{0}(\omega_{C}(-(g-i)p))$. Rescaling
$\alpha$ we can assume that $\alpha=(z^{g-i+1}+O(z^{g-i}))dz$. Adding to
$\alpha$ a linear combination of $\alpha[g-i],\ldots,\alpha[g-1]$ we get the
required element $\alpha[g-i+1]=(z^{g-i+1}+O(z^{g}))dz$. The uniqueness is
clear.

(ii) This easily follows from the fact that $h^{0}(\omega_{C}(np))=g-1+n$ for
$n\geq 2$ and from the fact that elements of $H^{0}(\omega_{C}(np))$ have zero
residue at $p$.

(iii) Let us start with an arbitrary formal parameter $z$, and construct a
basis of $H^{0}(\omega_{C})$ as above. Let

$\alpha[g-1]=(z^{g-1}+a_{g}z^{g}+\ldots)dz.$

We have to check that there is a unique series $u(z)=z+c_{2}z^{2}+\ldots$ such
that

$\alpha[g-1]=u(z)^{g-1}\cdot du(z)=u(z)^{g-1}u^{\prime}(z)dz,$

i.e.,

$(1+c_{2}z+c_{3}z^{2}+\ldots)^{g-1}\cdot(1+2c_{2}z+3c_{3}z^{2}+\ldots)=1+a_{g}z+a_{g+1}z^{2}+\ldots.$

We can solve this for $(c_{i})_{i\geq 2}$ step by step:

$(g+1)c_{2}=a_{g},\ \ (g+2)c_{3}+(g-1)(g-2)c_{2}^{2}=a_{g+1},\text{ etc.}$

∎

Let us consider the basis
$(\alpha[0],\ldots,\alpha[g-1],\omega[-2],\omega[-3],\ldots)$ of
$\omega_{C}(C\setminus p)$. Let us consider the expansions

$\alpha[i]=(z^{i}+z^{g}\sum_{n\geq 0}\alpha_{n}[i]z^{n})dz,\ \text{ for
}i=0,\ldots,g-1,$ $\omega[-i]=(z^{-i}+z^{g}\sum_{n\geq
0}\omega_{n}[-i]z^{n})dz,\ \text{ for }i\geq 2.$

Note that the special choice of a parameter $z$ as in Lemma 5.2(iii)
corresponds to $\alpha_{n}[g-1]=0$ for $n\geq 0$. From now on we assume that
$z$ is chosen in this way.

######  Lemma 5.3.

(i) For each $n\geq 1$, let $f[-g-n]$ denote the unique element of
$H^{0}(C,{\mathcal{O}}_{C}((g+n)p))$, up to additive constant, with
$f[g+n]=z^{-g-n}+O(z^{-g})$. Then up to an additive constant we have the
expansion

$\displaystyle-f[-g-n]+\frac{1}{z^{g+n}}=\frac{\alpha_{n-1}[g-2]}{z^{g-1}}+\frac{\alpha_{n-1}[g-3]}{z^{g-2}}+\ldots+\frac{\alpha_{n-1}[0]}{z}$
$\displaystyle+\omega_{n-1}[-2]\cdot z+\omega_{n-1}[-3]\cdot
z^{2}+\ldots+\omega_{n-1}[-g-1]\cdot z^{g}+O(z^{g+1}).$

(ii) For every $n\geq 1$, we have

$\displaystyle df[-g-n]+(g+n)\omega[-g-n-1]=$
$\displaystyle(g-1)\alpha_{n-1}[g-2]\omega[-g]+(g-2)\alpha_{n-1}[g-3]\omega[-g+1]+\ldots+\alpha_{n-1}[0]\omega[-2]$
$\displaystyle-\omega_{n-1}[-2]\alpha[0]-2\omega_{n-1}[-3]\alpha[1]-\ldots-g\omega_{n-1}[-g-1]\alpha[g-1].$

###### Proof.

(i) This follows from the residue theorem which gives equations

$\operatorname{Res}_{p}(f[-g-n]\cdot\alpha[i])=\operatorname{Res}_{p}(f[-g-n]\cdot\omega[-j])=0.$

(ii) Differentiating the expansion of $f[-g-n]$ found in part (i), we get

$\displaystyle\frac{df[-g-n]}{dz}=\frac{g+n}{z^{g+n+1}}+\frac{(g-1)\alpha_{n-1}[g-2]}{z^{g}}+\ldots+\frac{\alpha_{n-1}[0]}{z^{2}}$
$\displaystyle-\omega_{n-1}[-2]-2\omega_{n-1}[-3]z-\ldots-g\omega_{n-1}[-g-1]z^{g-1}+O(z^{g}),$

which implies that the difference between both sides in the claim identity
belongs to $H^{0}(C,\omega_{C}(-gp))=0$. ∎

######  Corollary 5.4.

The rational forms
$\alpha[0],\ldots,\alpha[g-1],\omega[-2],\ldots,\omega[-g-1]$ form a basis of
$H^{1}(C,\operatorname{\mathbb{C}})$, identified with the quotient of
$\omega_{C}(C\setminus p)/d{\mathcal{O}}(C\setminus p)$. For $n\geq 1$, we
have

$\displaystyle\omega[-g-n-1]\equiv\frac{g-1}{g+n}\alpha_{n-1}[g-2]\omega[-g]+\frac{g-2}{g+n}\alpha_{n-1}[g-3]\omega[-g+1]+\ldots+\frac{1}{g+n}\alpha_{n-1}[0]\omega[-2]$
$\displaystyle-\frac{1}{g+n}\omega_{n-1}[-2]\alpha[0]-\frac{2}{g+n}\omega_{n-1}[-3]\alpha[1]-\ldots-\frac{g}{g+n}\omega_{n-1}[-g-1]\alpha[g-1].$

###  5.3. Differentials near a separating node

Let us consider the extended clutching construction for a pair of curves with
formal parameters at marked points, $(C_{1},p_{1},x_{1})$,
$(C_{2},p_{2},x_{2})$, such that for $i=1,2$, genus of $C_{i}$ is $g_{i}\geq
1$ and $p_{i}$ is not a Weierstrass point on $C_{i}$, i.e.,
$H^{1}(C_{i},{\mathcal{O}}(g_{i}p_{i}))=0$. Here we assume that the formal
parameters $x_{i}$ are chosen as in Lemma 5.2(iii). Let ${\mathcal{C}}$ be the
corresponding family of curves over $S_{N}=\operatorname{Spec}(A_{N})$, where
$A_{N}=\operatorname{\mathbb{C}}[q]/(q^{n+1})$. We want to construct a special
basis of the bundle $\pi_{*}\omega_{{\mathcal{C}}/S_{N}}$ over $S_{N}$.

The sections of $\pi_{*}\omega_{{\mathcal{C}}/S_{N}}$ can be described by the
quadruples $(\omega_{1},\phi_{1}(x_{1}),\omega_{2},\phi_{2}(x_{2}))$,
$\omega_{i}$ being a global form on $(C_{i}\setminus\\{p_{i}\\})_{A_{N}}$ and
$\phi_{i}(x_{i})\in A_{N}[\\![x_{i}]\\!]$, such that

$\omega_{1}(x_{1})=\phi_{1}(x_{1})dx_{1}-q\phi_{2}(\frac{q}{x_{1}})\frac{dx_{1}}{x_{1}^{2}},\
\
\omega_{2}(x_{2})=\phi_{2}(x_{2})dx_{2}-q\phi_{1}(\frac{q}{x_{2}})\frac{dx_{2}}{x_{2}^{2}}.$
(5.3)

Actually, $\phi_{1}(x_{1})$ and $\phi_{2}(x_{2})$ are uniquely determined by
these equations from $\omega_{1}$ and $\omega_{2}$ (as regular parts in their
expansions).

We consider the differentials $\omega[-n]$ on $C_{1}$, regular outside
$p_{1}$, defined as in Sec. 5.2. Let us denote by
$\Lambda_{1}\subset\omega(C_{1}\setminus\\{p_{1}\\})$ the linear span of
$(\omega[-n])_{n\geq 2})$. Note that we have a direct sum decomposition

$\omega(C_{1}\setminus\\{p_{1}\\})=H^{0}(C_{1},\omega_{C_{1}})\oplus\Lambda_{1}.$

Similarly, we define the subspace
$\Lambda_{2}\subset\omega(C_{2}\setminus\\{p_{2}\\})$, where we denote the
basis in $\Lambda_{2}$ by $(\omega^{\prime}[-n])_{n\geq 2}$.

######  Proposition 5.5.

(i) For every regular differential $\alpha$ on $C_{1}$ there exists a unique
global section $\Phi_{1}(\alpha)$ of $\omega_{{\mathcal{C}}/S_{N}}$ with

$\omega_{1}\equiv\alpha\mod qA_{N}\otimes\Lambda_{1},\ \ \omega_{2}\equiv
0\mod qA_{N}\otimes\Lambda_{2}.$

This gives a linear embedding $\Phi_{1}:H^{0}(C_{1},\omega_{C_{1}})\to
H^{0}(\omega_{{\mathcal{C}}/S_{N}})$.

(ii) For $i=0,\ldots,g_{1}-1$, one has
$\Phi_{1}(\alpha[i])=(\omega_{1},\omega_{2})$ with

$\omega_{1}\equiv\alpha[i]+q^{i+g_{2}+2}\omega^{\prime}_{0}[-i-2]\cdot\omega[-g_{2}-2]+\ldots+q^{g_{1}+g_{2}+1}\omega^{\prime}_{g_{1}-i-1}[-i-2]\cdot\omega[i-g_{1}-g_{2}-1]\mod
q^{g_{1}+g_{2}+2},$
$\omega_{2}\equiv-q^{i+1}\omega^{\prime}[-i-2]-q^{g_{1}+1}\alpha_{0}[i]\cdot\omega^{\prime}[-g_{1}-2]-\ldots-q^{g_{1}+g_{2}+1}\alpha_{g_{2}}[i]\cdot\omega^{\prime}[-g_{1}-g_{2}-2].$

(iii) Similarly, we define a linear embedding
$\Phi_{2}:H^{0}(C_{2},\omega_{C_{2}})\to H^{0}(\omega_{{\mathcal{C}}/S_{N}})$.
The induced map

$(\Phi_{1},\Phi_{2}):H^{0}(C_{1},\omega_{C_{1}})\otimes A_{N}\oplus
H^{0}(C_{2},\omega_{C_{2}})\otimes A_{N}\to
H^{0}({\mathcal{C}},\omega_{{\mathcal{C}}/S_{N}})=H^{0}(S_{N},\pi_{*}\omega_{{\mathcal{C}}/S_{N}})$

is an isomorphism.

###### Proof.

(i) We solve the equations (5.3) order by order in $q$. Modulo $q$ we require
to have $\omega_{1}\equiv\alpha$, $\omega_{2}\equiv 0$, so
$\phi_{1}(x_{1})dx_{1}$ is the expansion of $\alpha$ at $p_{1}$ and
$\phi_{2}=0$. Assume now $N\geq 1$ and we already know
$(\omega_{1},\phi_{1},\omega_{2},\phi_{2})$ modulo $q^{N}$. Since
$\phi_{i}(x_{i})$ are regular, knowing $\phi_{i}$ modulo $q^{N}$ determines
uniquely polar parts of $\omega_{1}(x_{1})$ and $\omega_{2}(x_{2})$ modulo
$q^{N+1}$. More precisely, we have

$\omega_{1}(x_{1})_{\leq-1}\equiv-q\phi_{2}(\frac{q}{x_{1}})\frac{dx_{1}}{x_{1}^{2}}\mod(q^{N+1}),$

and similarly for the polar part of $\omega_{2}(x_{2})$. Since
$\phi_{2}(\frac{q}{x_{1}})\frac{dx_{1}}{x_{1}^{2}}\in\operatorname{span}(x_{1}^{-2},x_{1}^{-3},\ldots)$,
there exists a unique $\lambda_{1}\in qA_{N}\otimes\Lambda$ with

$(\lambda_{1})_{\leq-1}=-q\phi_{2}(\frac{q}{x_{1}})\frac{dx_{1}}{x_{1}^{2}}.$

Hence, the unique solution for $\omega_{1}$ is
$\omega_{1}=\alpha+\lambda_{1}$. The same argument gives a unique solution for
$\omega_{2}$. Now $\phi_{i}(x_{i})$ modulo $q^{N+1}$ is determined as the
regular part of the expansion of $\omega_{i}(x_{i})$:

$\phi_{i}(x_{i})dx_{i}=\omega_{1}(x_{1})_{\geq 0}.$

(ii) We just have to check that for these $\omega_{1}$, $\omega_{2}$ and for
$\phi_{i}(x_{i})dx_{i}=\omega_{i}(x_{i})_{\geq 0}$, $i=1,2$, equations (5.3)
are satisfied modulo $q^{g_{1}+g_{2}+2}$.

First, let us calculate $\phi_{1}(\frac{q}{x_{2}})$ modulo
$q^{g_{1}+g_{2}+1}$. Recall that $\omega[-g_{2}-2]_{\geq 0}=O(x_{1}^{g_{1}})$,
hence only the term $\alpha[i]$ in the formula for $\omega_{1}$ will
contribute:

$\phi_{1}(\frac{q}{x_{2}})\equiv\frac{q^{i}}{x_{2}^{i}}+\sum_{n\geq
0}\alpha_{n}[i]\frac{q^{n+g_{1}}}{x_{2}^{n+g_{1}}}\mod q^{g_{1}+g_{2}+1}.$

Hence, we see that $-q\phi_{1}(\frac{q}{x_{2}})dx_{2}/x_{2}^{2}$ matches the
polar part of $\omega_{2}$ modulo $q^{g_{1}+g_{2}+2}$.

Similarly, when we compute $\phi_{2}(\frac{q}{x_{1}})$ modulo
$q^{g_{1}+g_{2}+1}$, only the first term in the formula for $\omega_{2}$ will
contribute:

$\phi_{2}(\frac{q}{x_{1}})\equiv-\sum_{n\geq
0}\omega^{\prime}_{n}[-i-2]\frac{q^{n+g_{2}+i+1}}{x_{1}^{n+g_{2}}}\mod
q^{g_{1}+g_{2}+1}.$

Again this implies that $-q\phi_{2}(\frac{q}{x_{1}})dx_{1}/x_{1}^{2}$ matches
the polar part of $\omega_{1}$ modulo $q^{g_{1}+g_{2}+2}$.

(iii) Note that $H^{0}(S_{N},\pi_{*}\omega_{{\mathcal{C}}/S_{N}})$ is a free
$A_{N}$-module of a finite rank, and its reduction modulo $A_{N}/qA_{N}$ gets
identified with

$H^{0}(C_{0},\omega_{C_{0}})\simeq H^{0}(C_{1},\omega_{C_{1}})\oplus
H^{0}(C_{2},\omega_{C_{2}}).$

Now the assertion follows from the fact that our map $(\Phi_{1},\Phi_{2})$
reduces to the identity modulo $q$. ∎

Let

$\Pi:[H^{0}(C_{1},\omega_{C_{1}})\oplus H^{0}(C_{2},\omega_{C_{2}})]\otimes
A_{N}\to
H^{1}({\mathcal{C}},\pi^{-1}{\mathcal{O}}_{S_{N}})\simeq[H^{1}(C_{1},\operatorname{\mathbb{C}})\oplus
H^{1}(C_{2},\operatorname{\mathbb{C}})]\otimes A_{N}$

denote the composition of the isomorphism $(\Phi_{1},\Phi_{2})$ with the map
(5.2). We can expand $\Pi$ in powers of $q$:

$\Pi=\Pi_{0}+q\Pi_{1}+q^{2}\Pi_{2}+\ldots$

Note that $\Pi_{0}$ is just the sum of the standard embeddings
$H^{0}(C_{j},\omega_{C_{j}})\to H^{1}(C_{j},\operatorname{\mathbb{C}})$, over
$j=1,2$. From Proposition 5.5 we can calculate $\Pi_{j}$ for $j\leq
g_{1}+g_{2}+1$.

######  Corollary 5.6.

For $1\leq j\leq g_{1}+g_{2}+1$, and $0\leq i\leq g_{1}-1$, one has

$\Pi_{j}(\alpha[i])=(\omega^{\prime}_{j-i-
g_{2}-2}[-i-2]\omega[-j+i],-\delta_{j,i+1}\omega^{\prime}[-i-2]-\alpha^{\prime}_{j-g_{1}-1}[i]\cdot\omega^{\prime}[-j-1]),$

where the term with the negative index is considered to be zero, and the image
of $\omega[-n]$ in $H^{1}(C_{1},\operatorname{\mathbb{C}})$ for $n\geq
g_{1}+1$ is given by Corollary 5.4 (the image of $\omega^{\prime}[-n]$ in
$H^{1}(C_{2},\operatorname{\mathbb{C}})$ is given by similar formulas).

In particular, for $1\leq j\leq\min(g_{1},g_{2})$, one has

$\Pi_{j}(\alpha[j-1],0)=(0,-\omega^{\prime}[-j-1]),\ \
\Pi_{j}(0,\alpha^{\prime}[j-1])=(-\omega[-j-1],0)$

and $\Pi_{j}$ is zero on all other basis vectors.

Looking only at $\Pi_{1}$ we get the following description of the tangent map
to the period mapping, viewed as a map from the moduli space to the
Grassmannian. Note that if $C_{0}$ is the stable curve glued from $C_{1}$ and
$C_{2}$ as above (for $q=0$) then the period mapping sends $C_{0}$ to the
point of the Grassmannian corresponding to the subspace

$H^{0}(C_{1},\omega_{C_{1}})\oplus H^{0}(C_{2},\omega_{C_{2}})\subset
H^{1}(C_{1},\operatorname{\mathbb{C}})\oplus
H^{1}(C_{2},\operatorname{\mathbb{C}}).$

The tangent space to the Grassmannian at this point is identified with

$T_{C_{0}}G=\operatorname{Hom}_{\operatorname{\mathbb{C}}}(H^{0}(C_{1},\omega_{C_{1}})\oplus
H^{0}(C_{2},\omega_{C_{2}}),H^{1}(C_{1},{\mathcal{O}}_{C_{1}})\oplus
H^{1}(C_{2},{\mathcal{O}}_{C_{2}}).$

######  Proposition 5.7.

Let $(C_{1},p_{1},x_{1})$ (resp., $(C_{2},p_{2},x_{2})$) be a smooth curve of
genus $g_{1}\geq 1$ (resp., $g_{2}\geq 1$) with a marked point and a formal
parameter at it. Assume that $p_{1}$ (resp., $p_{2}$) is not a Weierstrass
point. Consider the functionals

$\phi_{i}:H^{0}(C_{i},\omega_{C_{i}})\to\omega_{C_{i}}|_{p_{i}}\simeq\operatorname{\mathbb{C}},i=1,2$

and the vectors

$\delta_{i}:\operatorname{\mathbb{C}}\simeq
H^{0}(C_{i},{\mathcal{O}}_{C_{i}}(p_{i})/{\mathcal{O}}_{C_{i}})\to
H^{1}(C_{i},{\mathcal{O}}_{C_{i}}).$

Here we use the trivialization of $\omega_{C_{i}}|_{p_{i}}$ (resp., of
${\mathcal{O}}_{C_{i}}(p_{i})/{\mathcal{O}}_{C_{i}}$) induced by $dx_{i}$
(resp., $1/x_{i}$). Then the derivative of the period map at $C_{0}=C_{1}\cup
C_{2}$ with respect to the parameter $q$ (coming from the extended clutching
construction) is given by

$(\delta_{2}\phi_{1},\delta_{1}\phi_{2})\in\operatorname{Hom}_{\operatorname{\mathbb{C}}}(H^{0}(C_{1},\omega_{C_{1}}),H^{1}(C_{2},{\mathcal{O}}_{C_{2}}))\oplus\operatorname{Hom}(H^{0}(C_{2},\omega_{C_{2}}),H^{1}(C_{1},{\mathcal{O}}_{C_{1}}))\subset
T_{C_{0}}G.$

###### Proof.

By Corollary 5.6, we have

$\Pi_{1}(\alpha[0],0)=(0,-\omega^{\prime}[-2]),\ \
\Pi_{1}(0,\alpha^{\prime}[0])=(-\omega[-2],0)$

and $\Pi_{1}$ is zero on all other basis vectors.

It is clear from the definition that $\phi_{1}$ is the functional on
$H^{0}(C_{i},{\mathcal{O}}_{C_{i}}(p_{i})$ sending $\alpha[i]$ with $i>0$ to
zero, and sending $\alpha[0]$ to $1$. It remains to check $\delta_{1}$ is the
image of $-\omega[-2]$ under the composition

$\omega_{C_{1}}(C\setminus\\{p_{1}\\})\to
H^{1}(C_{1},[{\mathcal{O}}_{C_{1}}\to\omega_{C_{1}}])\to
H^{1}(C_{1},{\mathcal{O}}_{C_{1}}).$

This map sends a differential $\eta$ to the class of the Laurent series
$\phi(x_{1})$ such that $\eta=d\phi(x_{1})$ in the formal neighborhood of
$p_{1}$. Since $-\omega[-2]=-\frac{1}{x_{1}^{2}}+\ldots$, we have in our case
$\phi(x_{1})=-\frac{1}{x_{1}}+O(1)$, which implies our statement. ∎

###  5.4. Separating node boundary in $g=2$ moduli

Now we specialize to the case $g_{1}=g_{2}=1$.

First of all, for $g=1$ we can express all objects in Sec. 5.2 in terms of
elliptic functions. Let
$C=\operatorname{\mathbb{C}}/(\operatorname{\mathbb{Z}}+\operatorname{\mathbb{Z}}\tau)$
be an elliptic curve with $0$ as the marked point. The canonical formal
parameter of Lemma 5.2(iii) is just the coordinate $z$ coming from the complex
plane, with $\alpha[0]=dz$.

The rational functions $f[-n]$, $n\geq 2$ (with a particular normalization of
additive constants), are expressed in terms of the rescaled derivatives of the
Weierstrass $\wp$-function:

$f[-n]=(-1)^{n}\frac{1}{(n-1)!}\wp^{(n-2)}(z)-\frac{c_{n-2}}{n-1}=\frac{1}{z^{n}}+(-1)^{n}\sum_{m\geq
1}{m+n-2\choose m}\frac{c_{m+n-2}}{n-1}z^{m},$

where

$\wp(z)=\frac{1}{z^{2}}+c_{2}z^{2}+c_{4}z^{4}+\ldots$

(where $c_{i}=c_{i}(\tau)$ and $c_{i}=0$ for odd $i$ and $c_{0}=0$). The above
normalization of $f[-n]$ satisfies $f[-n]=z^{-n}+O(z)$. Hence, the rational
differentials $\omega[-n]$ are given by

$\omega[-n]=f[-n]dz.$

The basis of $H^{1}(C,\operatorname{\mathbb{C}})\simeq\omega_{C}(C\setminus
p)/d{\mathcal{O}}(C\setminus p)$ is given by $\alpha[0]=dz$ and
$\omega[-2]=\wp(z)dz$. Note that for $n\geq 3$ we have

$\omega[-n]\equiv-\frac{c_{n-2}}{n-1}\cdot dz$

in $H^{1}(C,\operatorname{\mathbb{C}})$.

Now we consider the extended clutching applied to the elliptic curves
$C_{1}=\operatorname{\mathbb{C}}/(\operatorname{\mathbb{Z}}+\operatorname{\mathbb{Z}}\tau_{1})$
and
$C_{2}=\operatorname{\mathbb{C}}/(\operatorname{\mathbb{Z}}+\operatorname{\mathbb{Z}}\tau_{2})$
with the origins as the marked points.

We claim that $\Phi_{1}(dx_{1})$ can be found in the form

$\Phi_{1}(dz)=([1+\sum_{n\geq
1}a_{2n}(q)f[-2n](x_{1},\tau_{1})]dx_{1},-q[f[-2](x_{2},\tau_{2})+\sum_{n\geq
2}b_{2n}(q)f[-2n](x_{2},\tau_{2})]dx_{2}),$

for some series $a_{2n}(q)$, $n\geq 1$ and $b_{2n}(q)$, $n\geq 2$, depending
on $\tau_{1}$ and $\tau_{2}$. Indeed, we have

$\phi_{1}(x_{1})=1+\sum_{n\geq 1}a_{2n}(q)f[-2n](x_{1},\tau_{1})_{>0},\ \
\phi_{2}(x_{2})=-qf[-2](x_{2},\tau_{2})_{>0}-q\sum_{n\geq
2}b_{2n}(q)f[-2n](x_{2},\tau_{2})_{>0},$

and equations (5.3) can be rewritten as

$\sum_{n\geq
1}\frac{a_{2n}(q)}{x_{1}^{2n}}=-\frac{q}{x_{1}^{2}}\phi_{2}(\frac{q}{x_{1}})=\frac{q^{2}}{x_{1}^{2}}\bigl{(}f[-2](\frac{q}{x_{1}},\tau_{2})_{>0}+\sum_{n\geq
2}b_{2n}(q)f[-2n](\frac{q}{x_{1}},\tau_{2})_{>0}\bigr{)},$ $\sum_{n\geq
2}\frac{b_{2n}(q)}{x_{2}^{2n}}=\frac{1}{x_{2}^{2}}\sum_{n\geq
1}a_{2n}(q)f[-2n](\frac{q}{x_{2}},\tau_{1})_{>0}.$

These are equivalent to the following recursion for $(a_{2n})_{n\geq 1}$,
$(b_{2n})_{n\geq 2}$:

$a_{2n}=q^{2n}[c_{2n-2}(\tau_{2})+\sum_{m\geq 2}b_{2m}\cdot{2m+2n-4\choose
2n-2}\frac{c_{2m+2n-4}(\tau_{2})}{2m-1}],$ $b_{2n}=q^{2n-2}\sum_{m\geq
2}a_{2m}\cdot{2m+2n-4\choose 2n-2}\frac{c_{2m+2n-4}(\tau_{1})}{2m-1},\text{
for }n\geq 2,$

with the initial condition $a_{2}=0$. This recursion has a unique solution,
which proves our claim.

For example, we have

$a_{4}\equiv
c_{2}(\tau_{2})q^{4}+4c_{4}(\tau_{1})c_{2}(\tau_{2})c_{4}(\tau_{2})q^{10}\mod
q^{12},\ a_{6}\equiv c_{4}(\tau_{2})q^{6}\mod q^{12},\ a_{8}\equiv
c_{6}(\tau_{2})q^{8}\mod q^{12},$ $b_{4}\equiv
2c_{4}(\tau_{1})c_{2}(\tau_{2})q^{6}+3c_{6}(\tau_{1})c_{4}(\tau_{2})q^{8}\mod
q^{10},\ b_{6}\equiv 5c_{6}(\tau_{1})c_{2}(\tau_{2})q^{8}\mod q^{10},$

while all higher $a_{2n}$ (resp., $b_{2n}$) are zero modulo $q^{12}$ (resp.,
$q^{10}$).

The map $\Pi$ sends the differential $(dx_{1},0)$ to the class

$([1-\sum_{n\geq
2}a_{2n}(q)\frac{c_{2n-2}(\tau_{1})}{2n-1}]dx_{1},-q\omega[-2]+q[\sum_{n\geq
2}b_{2n}(q)\frac{c_{2n-2}(\tau_{2})}{2n-1}]dx_{2})$

in $(H^{1}(C_{1},\operatorname{\mathbb{C}})\oplus
H^{1}(C_{2},\operatorname{\mathbb{C}}))\otimes\operatorname{\mathbb{C}}[q]/(q^{N+1})$
(the formula for $\Pi(0,dx_{2})$ is obtained by switching the roles of $C_{1}$
and $C_{2}$).

Thus, modulo $q^{11}$ we have

$\displaystyle\Pi(dx_{1},0)\equiv$
$\displaystyle([1-\frac{c_{2}(\tau_{1})c_{2}(\tau_{2})}{3}q^{4}-\frac{c_{4}(\tau_{1})c_{4}(\tau_{2})}{5}q^{6}-\frac{c_{6}(\tau_{1})c_{6}(\tau_{2})}{7}q^{8}-\frac{4}{3}c_{2}(\tau_{1})c_{4}(\tau_{1})c_{2}(\tau_{2})c_{4}(\tau_{2})q^{10}]dx_{1},$
$\displaystyle-q\omega[-2]+[\frac{2}{3}c_{4}(\tau_{1})c_{2}(\tau_{2})^{2}q^{7}+2c_{6}(\tau_{1})c_{2}(\tau_{2})c_{4}(\tau_{2})q^{9}]dx_{2}).$

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