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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.

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  • $\begingroup$ I've also wondered this and would love to see a detailed answer. It is noticeably lacking from the stacks project. $\endgroup$ Nov 20 '20 at 17:54
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    $\begingroup$ 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. $\endgroup$ Nov 20 '20 at 18:38
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    $\begingroup$ 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. $\endgroup$ Nov 21 '20 at 18:49
  • $\begingroup$ @JasonStarr Do you mean a point specializes to $B\mathbb{G}_{m}$ (in, say $[\mathbb{A}^{1}/\mathbb{G}_{m}]$), etc? $\endgroup$ Nov 21 '20 at 21:13
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    $\begingroup$ 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. $\endgroup$
    – Angelo
    Nov 27 '20 at 17:26

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