# What is the local structure of a general Artin stack?

Let $$X$$ be an Artin stack over the complex numbers. What can one say about the local structure of $$X$$, i.e. what is the simplest class of stacks by which were can always find a cover of $$X$$ by open substacks in this class? For example, does $$X$$ always have a cover by open substacks of the form $$U/G$$ where $$G$$ is an algebraic group acting on a scheme or complex analytic space $$U$$? I actually suspect that this is false, and that a counterexample is given by the moduli stack of nodal genus zero curves (I don't see how to construct such a neighborhood around $$\mathbb CP^1\vee\mathbb CP^1$$). If it is indeed false, what is the best one can say? I am very willing to pass to the analytification of $$X$$ if this helps get a stronger local structure result.

A related question: Stacks as local quotients or via atlases

• What do you mean by the "coarse space"? – Angelo Nov 23 '19 at 14:25
• @Angelo: Thank you, I've rephrased the question to avoid that notion. – John Pardon Nov 23 '19 at 15:50
• A DM stack has a coarse space, but it's still an algebraic space, and algebraic spaces can be pathological. – Ben Wieland Nov 23 '19 at 20:36
• Maybe I don't get the question, but if your stack is etale locally a quotient stack then surely it is also locally a quotient stack for the analytic topology? Since an etale cover can be refined to an analytic open cover. – Dan Petersen Nov 24 '19 at 5:40
• A stack which is the quotient by a group $G$ has inertia all subgroups of $G$. A finite dimensional algebraic group cannot contain infinitely many abelian varieties (up to isogeny?). Thus the moduli of (unmarked) genus 1 curves, which has inertia all elliptic curves is not locally a quotient stack. But I think that a finite dimesional family of unipotent groups can be embedded in a single linear group. – Ben Wieland Nov 25 '19 at 2:19