Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that

- $f$ is proper, flat, and of finite presentation;
- The geometric fibers of $f$ are reduced, connected, one-dimensional, of arithmetic genus $g$, and smooth away from finitely many points at which the singularities are required to be ordinary double points.

Does $f$, Zariski locally on $S$, come as a base change of a semistable curve $f' \colon X' \rightarrow S'$ with $S'$ regular?

A lemma of this sort seems to be used in the proof of 9.4/1 of "Neron models," i.e., in the proof of the representability of $\mathrm{Pic}^0_{X/S}$ by a scheme. The authors cite the paper of Deligne and Mumford for this reduction to a regular base. I can't see how the citation justifies the claim because Deligne and Mumford seem to deal with a narrower class of curves, namely, they impose an additional condition:

- If a geometric fiber of $f$ has an irreducible component isomorphic to $\mathbb{P}^1$, then that component meets the other components in $\ge 3$ points.

EDIT. The real question is: for a semistable curve $f$ as above, is $\mathrm{Pic}^0_{X/S}$ a scheme, as claimed in 9.4/1 of "Neron models"? The proof given there seems incomplete (see the comments below for some discussion). Any ideas are welcome.

etale-locallyon the base. But any semistable curve becomes a stable "marked" curve etale-locally on the base, so one can use the $\mathbf{Z}$-smooth Deligne-Mumford stack of stable $n$-pointed genus-$g$ curves for suitable $n$ with $2g-2+n > 0$ (studied in a paper of Knudsen, building on the work of Deligne and Mumford). So this renders the likely erroneous reference to Deligne-Mumford in BLR moot. $\endgroup$ – user74230 Jan 26 '15 at 3:435more comments