The converse statement holds. If every atlas ( U ) of a smooth finite type separated DM-stack ( X ) over ( \mathbb{C} ) (with finite inertia) is a scheme, then the coarse moduli space ( X^c ) must be a scheme.

Key Reasoning:

1. Definition of Atlas and Coarse Space: An atlas of ( X ) is an étale surjective morphism ( U \to X ) where ( U ) is an algebraic space. The coarse mo...

Yes. In fact you don’t need the hypothesis “every atlas is a scheme”; the existence of one étale atlas by a scheme already forces the coarse moduli space to be a scheme.

Here is a convenient way to see it.

• Let π: X → Y be the coarse moduli space of a separated DM stack X with finite inertia (so Y is an algebraic space).
• Pick an étale atlas p: U → X with U a scheme (such an atlas always exist...

Yes – in the situation you describe the coarse moduli space is always a scheme, so the “converse” is automatically true. No counter–example can occur as long as

1. the total space P is quasi-projective over ℂ, and
2. the acting group G is a (connected) reductive algebraic group (or, more generally, a linearly reductive group scheme of finite type over ℂ).

Because of these two hypotheses ...

Yes – under the hypotheses you stated the coarse moduli space is always a scheme, so the converse of your theorem holds vacuously. In fact the property that “every atlas is a scheme’’ is not needed: for a smooth quasi-projective scheme P endowed with an action of a (smooth, finite type, separated) reductive group G over ℂ the quotient sheaf P // G = P/G = X^c is a quasi-projective (...

Coarse Moduli Spaces of Stacks for Which Every Atlas is a Scheme

This is an interesting question in algebraic stack theory. Let me provide a detailed analysis.

The Question's Framework

We have a smooth, finite type, separated DM-stack over with finite inertia, where:

• is a smooth quasi-projective scheme
• is a smooth reductive group scheme

The author has ...

This is an excellent question. The answer is indeed negative, as you suspected. We can construct a counterexample using the idea that the "stackiness" of can be concentrated on a small locus, forcing any atlas (which must be an algebraic space) to live on the part of the stack that behaves like a scheme.

The Strategy

The core idea is to construct a Deligne-Mumford stack such that: 1....

This is a very interesting question about the relationship between the nature of atlases of a stack and its coarse moduli space.

The answer to your question is negative - there exist stacks where every atlas is a scheme, but the coarse moduli space is not a scheme.

Here's a counterexample:

Counterexample: Consider the quotient stack where acts...

The short answer is: In fact nothing “mystical’’ is hiding here – already for any stack of the form

X = [P/G]  

with P smooth quasi–projective and G a reductive group over C (and with finite stabilizers) the Keel–Mori coarse moduli X^c is automatically a genuine scheme. In particular one does not need the extra hypothesis “every atlas of X is a scheme’’ in order to force X^c to be a schem...