I have 2 questions about the theorem III.1.2 of Deligne-Rapoport’s “Les shemas de modules de courbes elliptiques”.

1.

Let $k$ be a field, $Lambda$ a complete noetherian local ring with the residue field $k$, $C_0$ an elliptic curve or the Neron $N$-gon ($N$ relatively prime to the characteristic of $k$) with the standard generalized elliptic curve structure over $k$.

Then the deformation functor $D$ of $C_0$ (as a generalized elliptic curve, i.e., considering its morphism $m : C_0^text{sm} times C^0 to C^0$) on $mathscr{C}_Lambda$ is pro-representable.

First, the authors show that $C_0$ has no infinitisimal automorphisms over $k(epsilon)/epsilon^2$.

From this they conclude this highlighted statement.

I don’t understand why we get the pro-representability from this.

It seems that we can apply the arguments of (3.7) of Schelessinger’s “Functors of artin rings” even in this situation.

If so, then by this and by (3.8) of this Schlessinger’s paper we have the pro-representability.

But I don’t understand why we can apply the argument of (3.7), even if we add a morphism $C_0^text{sm} times C^0 to C^0$.

2.

Let $k$ be a field, $(C_0, e)$ the Neron 1-gon with the “identity section” over $k$ or an elliptic curve with its identity.

Let $A, A’$ be artin local rings with the residue fields $k$ and $A’ to A$ a small extension.

(i.e., a surjection whose kernel is annihilated by the maximal ideal of $A’$ and is 1-dimension over $k$.)

Consider $(C, e)$, where $C$ is proper flat scheme over $A$, $e in C^text{sm}(A)$, and $C times_A k cong C_0$, compatible with $e$.

Then the obstruction to deformation of this $(C, e)$ to $A’$ is in $operatorname{Ext}^2(Omega_{C_0 / k}(e), mathscr{O}_{C_0})$,

and its first order deformation is parametrized by $operatorname{Ext}^1(Omega_{C_0 / k}(e), mathscr{O}_{C_0})$.

Since $C_0/k$ is locally complete intersection, I know that the obstruction to deformation of $C$ (just as a scheme) to $A’$ is in $operatorname{Ext}^2(Omega_{C_0 / k}, mathscr{O}_{C_0})$, and its first order deformation is parametrized by $operatorname{Ext}^1(Omega_{C_0 / k}, mathscr{O}_{C_0})$.

(By the section 10 of Hartshorne’s “Deformation theory”.)

(This, for a stable curve of genus $ge 2$, is also described in Deligne-Mumford.)

Why do we need $Omega_{C_0 / k}(e)$ instead of $Omega_{C_0 / k}$, if we add a section?

And I think that the obstruction is in $operatorname{Ext}^2(Omega_{C_0 / k}, mathscr{O}_{C_0})$ even in this case.

Indeed, since $e$ is in $C^text{sm}(A)$, if there exists a deformation $C’$ of $C$ to $A’$, then by the formally smoothness, $e in C’^text{sm}(A)$ lifts to $e’ in C’^text{sm}(A’)$.

So the existence of the deformation of $C$ automatically induces the existence of the deformation of the section.

This argument is described in the proof of (2.2.2.4) of Kai-Wen Lan’s “Arithmetic Compactifications of PEL-Type Shimura Varieties”, to show the formally smoothness of the local moduli of abelian varieties.

This post is related to this, but it has no answer.