# On the local structure of Deligne-Mumford stacks

Is it true that for any DM stack $\mathcal{X}$ (quasicompact, separated, of finite type over a field) there is a Zariski covering $\mathcal{U}_i \to \mathcal{X}$ by open substacks, such that for all $i$ there is an etale finite surjective morphism $\phi_i: Z_i \to \mathcal{U}_i$, where $Z_i$ is a scheme? (Of course it will be true if we remove the condition that the morphisms $\phi_i$ must be finite.)

Upd. Oh, I think by Theorem 6.1 of Laumon, Moret-Bailly "Champs algebriques" it is equivalent to saying that $\mathcal{X}$ admits a Zariski covering by quotient stacks.

-
Something weaker is true: Etale locally on the coarse moduli space $\mathcal{X}$ is a quotient stack. See, for example, Theorem 2.12 in ma.utexas.edu/users/molsson/homstackfinal.pdf – Lennart Meier Sep 8 '13 at 1:28
Of course I know about it, thanks. – Nullstellensatz Sep 8 '13 at 10:51

## 1 Answer

Proposition 5.2 of https://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf has a partial answer to you question. If X is a DM stack whose coarse module space is a scheme, then it is Zariski locally a quotient stack if and only if it is Zariski locally a finite group quotient.

-