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Let $X$ be an algebraic stack of finite type over a field.

Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$?

By intrinsic here I mean using constructions such as the inertia stack, the stabilizer group construction, etc.

A natural conjecture is that this minimum should be something like the dimension of $X$ plus the dimension of the largest stabilizer of any point of $X$, as one can see using classifying stacks, for example.

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This is false: $B\mu_p$ in characteristic $p$ is an algebraic stack but not a DM stack (and in particular, does not have a 0-dimensional smooth cover, because such a cover would be etale).

(Note that while $B\mu_p$ is a point mod $\mu_p$, the map from the point to $B\mu_p$ is a $\mu_p$ torsor, and thus not etale since $\mu_p$ is not etale. It does have a cover by $\mathbb{G}_m$.)

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    $\begingroup$ Maybe one could define a smooth envelope for a group scheme as a smooth group scheme into which it embeds, so that the smooth envelope of $\mu_p$ is $\mathbb{G}_m$? Then it might be the dimension of $X$ plus the largest dimension of a smooth envelope of a stabilizer. $\endgroup$
    – David Corwin
    Commented Jan 10, 2023 at 10:46
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Professor Jarod Alper provided me an answer to this question in separate correspondence. When the stabilizer at a point of the stack is smooth, my conjecture above is true by the theory of miniversal deformations. This is Theorem 3.6.1 in Professor Alper's notes here: https://sites.math.washington.edu/~jarod/moduli.pdf.

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