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 $\mathrm{Ext}^2(\Omega_{C_0}(e),\mathcal{O}_{C_0})$.

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