  We show that various natural algebro-geometric moduli stacks, including the
stack of curves, have the property that every Deligne-Mumford gerbe over a
field appears as the residual gerbe of one of their points. These gerbes are
universal obstructions for objects of the stack to be defined over their fields
of moduli, and for the corresponding coarse moduli space to be fine. Thus, our
results show that many natural moduli stacks hold objects that are obstructed
from being defined over their fields of moduli in every possible way, and have
coarse spaces which fail to be fine moduli spaces in every possible way. A
basic insight enabling our arguments is that many classical constructions in
equivariant projective geometry generalize to the setting of relative geometry
over an arbitrary Deligne-Mumford gerbe over a field.
