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Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. (A good (but not equivalent) way to think about a stack whose good moduli space is a single point is that for any two points, the closure of their orbits intersect.)

One says that a stack $\mathcal X$ is a quotient stack if it is the quotient of an algebraic space by a subgroup of GL_n (see for instance EHKV, which also gives a criterion for a stack to be a quotient stack in terms of vector bundles).

One says that a stack has the resolution property if every coherent sheaf is a quotient of some vector bundle; Totaro's paper The resolution property for schemes and stacks relates the property of being a quotient stack to the resolution property.

Question: Let $\mathcal X$ be a stack with a good moduli space $\mathcal X \to X$ such that X is a geometric point (i.e., X = Spec k, where k is a separably closed field). Suppose further that the stabilizers are affine linearly reductive groups. Is $\mathcal X$ a quotient stack?

(See this answer for the definition of stabilizer of a point of a stack that isn't a quotient stack.)

Remarks:

  1. The condition on stabilizers excludes things like BE with E an elliptic curve.

  2. The condition that k is separably closed excludes non-trivial gerbes.

  3. I'd be just as happy with an answer to "Does the resolution property hold for $\mathcal X$ ?".

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    $\begingroup$ This is an interesting question. Here is a specific instance: let $n$ be a positive integer, and let $\mathcal X_n$ be the stack over $\mathbb C$ of nodal proper curves of genus 0 with at most $n$ nodes. One shows that it satisfies the conditions above. It is a quotient stack for $n \leq 1$, but I don't know the answer for any $n \geq 2$. $\endgroup$ – Angelo Feb 28 '11 at 7:03
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    $\begingroup$ The stack $\mathcal{X}_n$ is not a global quotient stack for $n\geq 2$ (Kresch: Flattening stratification and the stack of partial stabilisations of prestable curves). But, as noted by David Zureick-Brown in arxiv:1208.2882, $\mathcal{X}_n\to \mathrm{Spec}(\mathbb{C})$ is not a good moduli space so this does not answer the question. $\endgroup$ – David Rydh Feb 5 '13 at 19:10
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This is answered in general by Theorem 13.1 in our paper The étale local structure of algebraic stacks (arXiv:1912.06162). Let $\mathcal{X}$ be a stack with a good moduli space $X$ such that

  1. $\mathcal{X}$ has affine stabilizers,
  2. $\mathcal{X}$ has separated diagonal, and
  3. $\mathcal{X}$ is of finite presentation over an algebraic space.

Then $\mathcal{X}$ has the resolution property étale-locally (and even Nisnevich-locally) on $X$. In particular, if $X$ is the spectrum of a field or a henselian local ring, then $\mathcal{X}$ has the resolution property.

Conditions 1 and 2 are necessary. Note that having the resolution property locally on $X$ implies that $\mathcal{X}$ has affine diagonal.

In general $\mathcal{X}$ does not have the resolution property Zariski-locally on $X$. There is an example in SGA 3, Exp X, §1.6 (cf. Remark 2.5 in our paper), of a 2-dimensional torus $G$ over the nodal cubic curve $C$ such that $G$ is not locally isotrivial. This means that $G$ cannot be embedded in $\mathrm{GL}_N$ for any $N$, not even Zariski-locally on $C$. It follows that $\mathcal{X}=BG$ does not have the resolution property Zariski-locally on its good moduli space $X=C$.

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