Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)

Question

We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor.

And now we only consider the case that $C_0$ is irreducible as in D-R (Deligne-Rapoport) Chapter III, 1.3. Then $D$ is equivalent to the deformation functor $D'$ of the couple $(C_0,e)$, with $e$ a marked smooth point ($e$ corresponds to the identity of the law $+$). In D-R, it says that $\mathrm{Ext}^1(\Omega_{C_0}(e),\mathcal{O}_{C_0})$ is the space of first order deformations and the obstruction for infinitesimal lifting lives in $\mathrm{Ext}^2(\Omega_{C_0}(e),\mathcal{O}_{C_0})$.

1. Could someone explain why $\Omega_{C_0}(e)$ plays such a role? Or maybe you could point out the reference to me.

2. Another related question is: we know from D-R, $D$ is prorepresented by $(R,\xi)$ where $R$ is complete noetherian $\Lambda$-algebra with residue field $k$ and $\xi=\varprojlim \xi_n$ with $\xi_n\in D(R/m_R^{n+1})$, $\xi$ gives rise to a formal scheme $\mathcal{C}=\varinjlim C_n$ over $\mathrm{Spf} R$ with $C_n$ corresponding to $\xi_n$. Since $C_0$ is projective, Grothendieck's formal existence theorem garantees a unique projective curve $C$ over $\mathrm{Spec} R$ whose formal completion is $\mathcal{C}$.

   My question is why $C$ admits a structure of generalised elliptic curve? Or equivalently why the 'group' law algebraizes?

3. I should read D-R carefully. Question 2 follows from D-R Chapter II Prop. 1.5, Prop. 2.7. However, I have a similar question for toroidal models of abelian varieties, see Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties
   Remark: I say "similar" because both are about lifting group action from formal world to algebraic world.

Answer 1

Concerning #1, the twist on the sheaf is keeping track of the deformation of the identity section.

Concerning #2, in the residually reducible case a deformation of the underlying pointed curve need not admit any group scheme structure at all, due to non-homgeneity in the deformations at singularities (see Example 2.1.11 in the paper "Arithmetic Moduli of Generalized Elliptic Curves"). So the algebraization issue is more serious in the residually reducible case: one needs some extra "proper" data beyond geometry of a pointed curve to encode the group structure in a suitable manner.

Yet it can be achieved, as the data for the group law can be encoded in terms of "proper structures" via II, 3.2 in the D-R paper (that D-R use nowhere but is super-useful for this purpose). See Theorem 2.2.2, Cor. 2.2.3 and especially Cor. 2.2.4 in the above-mentioned paper for the existence/uniqueness of the group structure under deformation in the possibly residually reducible case via specifying "ample level structure" in an appropriate sense (which collapses to just the data of the identity section in the residually irreducible case); Cor. 2.2.4 in that paper makes precise the sense in which this is really a statement about deformations as generalized elliptic curves and does not require making a base change to attain a special level structure (and works over non-local bases subject to a mild necessary hypothesis on the geometry of reducible fibers, as one needs for arithmetic study along the cusps).

Concerning #3, perhaps look at work of Martin Olsson (or email Olsson).