Let $\mathscr{M}_*$ be the stack over $\mathbb{Z}$ which classifies generalized elliptic curves $E/S$, such that for every geometric point $k$ of $\mathbb{Z}$, the fibre $E_k$ is smooth or $n$-gon, where $\mathrm{char} k$ does not divide $n$. (see III.0. of Deligne-Rapoport's "Les schemas de modules de courbes elliptiques")
In III.2.5., the authors claim that $\mathscr{M}_*$ is locally of finite presentation as a fibred category, i.e., for any direct limit of rings $A = \lim A_i$, the canonical map $\lim \mathscr{M}_* (A_i) \to \mathscr{M}_*(A)$ is an equivalence. They say that this is EGA IV.8.8.
Using it (and some propositions in Stack Project), we can show, for a generalized elliptic curve $E/A$, that there exist an $i$, a proper flat curve $E_i / A_i$, and morphism $E_i^\text{sm} \times E_i \to E_i$, which induce the ones of $E/A$. But I can't show that for some $i$, this $E_i/A_i$ is a generalized elliptic cuvre.
If for some $i$ $E_i/A_i$ has reduced geometric fibres, then we can show that this $E_i/A_i$ has connected geometric fibres.
And it seems that I can show for some $i$, every geometric fibre of $E_i/A_i$ has the trivial dualizing sheaf.
(First assume $A_i$ noether. Then $\operatorname{Spce} A_i$ has an connected open subset which intersects the image of $\operatorname{Spec}A$.
Thus using the Picard scheme of $E_i \times U / U$, we can show the subset "$\{\omega = \mathscr{O} \}$" of $U$ is clopen.)
So finally, I want to show that for some $i$, every geometric fibre of $E_i/A_i$ is reduced, and whose singularity is at least ordinary double. (If so, then by the argument above and II.1.13., we have that this $E_i/A_i$ is a generalized elliptic curve.)
I can't find it in EGA. So please suggest me some references of it, or please prove it.
