# Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $$X$$ has an induced topological space $$|X|$$ given by equivalence classes of fields mapping to $$X$$ as outlined in the stacks project. If $$|X|$$ is connected, does that imply there exists a smooth map $$V \to X$$ from a connected scheme $$V$$?

My naive guess is to take an atlas $$V \to X$$ and try to glue it along its intersection, piecing together disjoint components of $$V$$, or perhaps take a colimit along inclusions over connected, smooth $$V \to X$$.

A positive answer to this question would simplify a technical argument.

• I've also wondered this and would love to see a detailed answer. It is noticeably lacking from the stacks project. Nov 20, 2020 at 17:54
• I like this question. It's a bit similar to asking: if $X$ is a connected scheme, does it admit a (smooth) surjection from a connected affine scheme? True if $X$ is qcqs with an ample family of line bundles, by Thomason's version of the Jouanolou trick. Nov 20, 2020 at 18:38
• That is definitely false without a qcqs hypothesis: a point is a specialization of $B\mathbb{G}_m$, in turn a specialization of $B\mathbb{G}_m^2$, etc. ad infinitum. Nov 21, 2020 at 18:49
• @JasonStarr Do you mean a point specializes to $B\mathbb{G}_{m}$ (in, say $[\mathbb{A}^{1}/\mathbb{G}_{m}]$), etc? Nov 21, 2020 at 21:13
• For Deligne-Mumford stacks, this is claimed as Proposition 4.4 in the original paper of Deligne-Mumford "On the irreducibility of the moduli space of curves", but without proof. I have never been able to figure out why this should be true. Nov 27, 2020 at 17:26