TITLE: Schematic locus of algebraic spaces in fibers
QUESTION [10 upvotes]: Let $S$ be a scheme and let $X$ be a quasi-separated algebraic space over $S$. Does there exist an open subspace of $X$ which is a scheme and which is dense in each fiber $X_s$, $s \in S$?
I am happy to make the following additional assumptions:
(1) $S$ is the spectrum of a complete dvr with algebraically closed residue field and $X$ is flat and of finite type over $S$.
(2) All fibers of $X$ are geometrically irreducible.
(3) The generic fiber of $X$ is geometrically normal and the special fiber of $X$ is geometrically integral (then $X$ is normal).

REPLY [6 votes]: A counter example to the question in the presence of (1), (2), and (3) is to take $\mathbf{G}_{m, S}$ and divide out by the etale equivalence relation given by $(x, s) \sim (x, s)$ and for $s \not = 0$ also $(x, s) \sim (-x, s)$ (assume the residue characteristic of your dvr is not $2$). Then the resulting morphisms
$$
\mathbf{G}_{m, S} \longrightarrow X \longrightarrow S
$$
have the following properties: the first is surjective \'etale and the second is smooth of relative dimension 1 and both $\mathbf{G}_{m, S}$ and $X$ have geometrically integral fibres over $S$. The schematic locus of X is exactly the complement of the special fibre. It seems to me the "problem" is that the degree of the function field extensions induced between the fibres of $\mathbf{G}_{m, S}$ and $X$ drops from $2$ to $1$ from the generic to the special fibre.
PS: As Jason points out on locally Noetherian separated algebraic spaces the schematic locus contains all codimension $1$ points, see Tag 0ADD