Murphy's Law for Algebraic Stacks

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.