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In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\mathbb C$-object up to isomorphism. Unless I'm not mistaken, it is not hard to see that $\mathcal X$ is isomorphic to $BG$ for some affine finite type (smooth) group scheme $G$ over $\mathbb C$.

On the other hand, it seems natural to wonder about the following "exercise":

What is the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely two $\mathbb C$-objects up to isomorphism?

It could be the disjoint union of $BG$ with another $BG'$, but it could also be a connected stack such as $[\mathbb A^1/\mathbb G_m]$. Are these the only possibilities up to isomorphism? Or is there more?

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    $\begingroup$ What do you mean by "such as"? There are many smooth varieties with an action of a linear algebraic group having precisely two orbits. $\endgroup$
    – Jason Starr
    Commented Feb 16, 2016 at 19:49
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    $\begingroup$ You can say something if you add a couple of additional hypotheses to your $\mathcal X$. For instance, if $\mathcal X$ has a dense open non-stacky $\mathbb C$-point and has reductive stabilizers, then $\mathcal X$ is isomorphic to $[X/T]$ with $X$ an affine toric variety and $T$ its torus. (The hypotheses on $\mathcal X$ are equivalent to saying that $\mathcal X$ is generically an algebraic space and that the unique stacky point has a reductive stabilizer.) This is a consequence of Theorem 4.5 of Geraschenko-Satriano's Toric Stacks II; see arxiv.org/abs/1107.1907 $\endgroup$
    – Ariyan Javanpeykar
    Commented Feb 17, 2016 at 9:11
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    $\begingroup$ I think that an $\mathcal X$ as in the question is isomorphic to a gerbe over $[\mathbb A^1/\mathbb G_m]$. Indeed, if the generic stabilizer is trivial and the geometric stabilizers are reductive, then one can use (up to checking the details) Geraschenko-Satriano's main theorem to see that $\mathcal X$ is isomorphic to $[\mathbb A^1/\mathbb G_m]$. $\endgroup$
    – Ariyan Javanpeykar
    Commented Feb 17, 2016 at 18:26

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Here's what you can say (following my comments above).

Lemma. Let $X$ be a smooth finite type connected algebraic stack over $\mathbb C$ with precisely two $\mathbb C$-objects (up to isomorphism). Suppose that

1) the diagonal of $X$ is affine;

2) the stabilizers of all geometric points of $X$ are reductive; and

3) the stack $X$ has a dense open non-stacky point.

Then $X\cong [\mathbb A^1/\mathbb G_m]$.

Proof. This follows from the proof of the main theorem of Geraschenko-Satriano given in Toric Stacks II; see arxiv.org/abs/1107.1907. Indeed, under our assumptions, $X$ has precisely one divisor. The natural open immersion associated to this Cartier divisor $X\to [\mathbb A^1/\mathbb G_m]$ is shown to be an isomorphism in their proof by using the local structure theorem of their paper (Theorem 4.5 in loc. cit.). QED

Remark. $[\mathbb P^1/\mathbb G_a]$ has precisely two objects, and it's not isomorphic to $[\mathbb A^1/\mathbb G_m]$ because condition 2) fails.

Remark. If you relax condition 3, then you can use "rigidification" to see that $X$ is a gerbe over $[\mathbb A^1/\mathbb G_m]$ for some reductive group $G$.

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