Homogenous varieties under a connected reductive group are another class of examples.

This is clearly false for finite groups (take the union of the two coordinate axes in $\mathbb A^2$, with the involution that switches the two axes. For a connected example, embed the cyclic group into $\mathbb G_\mathrm{m}$, and consider the induced action.

It is easy to give examples of smooth proper hyperbolic DM stacks, with plenty of effective pluricanonical divisors, whose moduli spaces are rational (this already occurs in dimension 1).

I apologize for the typo, and for not reading the post carefully enough.

Suppose that $y$ is a point of $Y$, and $X$ is the disjoint union of $Y$ and $\{y\}$.

There is no such flat map; the special fibers of $Pic_{Q/U} \to U$ are concentrated on proper subvarieties of $U$.

Formation of the Picard scheme commutes with base change, so the fibers of $Pic_{Q/U} \to U$ are complicated.

Minseon, you are right, sorry.

Anyway, it's a really nice argument.

Jason, does this work for fields that are not algebraically closed?

Consider the case that $S$ is the spectrum of a DVR, and $X$ the spectrum of its field of fractions.

What kind of sheaves do you have in mind? What's the "trivial" sheaf?

I think that the $\mathcal O$ notation comes from the notation for rings of integers in number fields, probably standing for "order" ("Ordnung" in German).

I see. This seems quite subtle.

If you take sufficiently many points in general position, shouldn't the condition be satisfied?