ag.algebraic geometry - Conditions for "bootstrapping" a smooth DM stack?

Notation: Let $x$ be a point of a smooth separated finite type DM stack $mathcal X$ over a field. Suppose
• $G$ is the stabilizer of $x$,
• $V$ is the tangent space of $x$ (which comes equipped with an action of $G$),
• $G^textrm{triv}subseteq G$ is the subgroup which acts trivially on $V$,
• $H = G/G^textrm{triv}$ (note $H$ acts on $V$),
• $K$ is the subgroup of $H$ generated by pseudoreflections on $V$, and
• $K'$ the commutator subgroup of $K$.

$mathcal X$ can be expressed as you described (in an étale neighborhood of $x$) if and only if $K'$ is trivial.

I'll now unpack that answer. Any smooth separated finite type DM stack over a field can be (canonically!) obtained from its coarse space with the following steps (this is basically the main Theorem of my paper with Matt Satriano, A "bottom up" characterization of smooth Deligne-Mumford stacks):

1. take the canonical stack of the coarse space,


2. do a root stack construction along the ramification divisor of the coarse space map, rooting each component of the ramification divisor by the degree of ramification,


3. take the canonical stack again (the root stack may not be smooth any more, but it will have quotient singularities!), and


4. add a gerbe.

Your question is "when can we skip step 3?" That is, when is the root stack from step 2 already smooth?

Using the notation above, and looking formally locally around $x$ (so we can assume $mathcal X=[V/G]$; you can describe it étale locally too, but it's clearer this way), the above steps are:

1. $bigl[(V/K)/(H/K)bigr]$ is the canonical stack of the coarse space $V/H = V/G$,


2. $bigl[(V/K')/(H/K')bigr]$ is a root stack of $bigl[(V/K)/(H/K)bigr]$,


3. $[V/H]$ is the canonical stack of $bigl[(V/K')/(H/K')bigr]$, and


4. $mathcal X = [V/G]$ is a $G^textrm{triv}$-gerbe over $[V/H]$.

Note that step 1 is the familiar way of building the canonical stack of a space with quotient singularities (in this case $V/H$) by expressing it as a quotient by a finite group somehow, and then quotienting out the subgroup generated by pseudoreflections, with the Chevalley-Shephard-Todd theorem ensuring that you don't lose smoothness. This description tells us that the canonical stack is any description of the space as a quotient where the group acts without pseudoreflections.

Note that in step 3, we're quotienting out by $K'$, which has no pseudoreflections since it's a commutator subgroup (all its elements must therefore act with determinant 1, and there are no pseudoreflections of determinant 1). So it makes sense that this step is a canonical stack.

It's pretty clear that step 4 is a (trivial) gerbe.

Seeing that step 2 is a root stack is more complicated; you can find the details in the section titled "A local description of Theorem 1" in the paper linked above.