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Let $X$ and $Y$ be K3 surfaces over the complex numbers.

Under what assumptions, do there exist

  1. a finite group $G_X$
  2. a finite group $G_Y$
  3. a $G_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology)
  4. a $G_Y$-gerbe $\mathcal{Y}\to Y$
  5. an isomorphism $\mathcal{X}\cong \mathcal{Y}$ (of algebraic stacks)?
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    $\begingroup$ In that case, $X$ is the coarse moduli space of $\mathcal{X}$, and $Y$ is the coarse moduli space of $\mathcal{Y}$. Thus, if $\mathcal{X}$ is isomorphic to $\mathcal{Y}$ as a stack in groupoids over the category of $\mathbb{C}$-schemes, then $X$ is isomorphic to $Y$ as $\mathbb{C}$-schemes. $\endgroup$
    – Jason Starr
    Jan 17, 2018 at 10:03
  • $\begingroup$ @JasonStarr Ok. This was arguably not a very intelligent question. Can you post your comment as an answer please? $\endgroup$
    – Neeroen123
    Jan 18, 2018 at 16:36

1 Answer 1

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As Jason Starr says in his comments, such data exists if and only if $X$ is isomorphic to $Y$.

Indeed, let $\mathcal{X}\to X$ be a $G_X$-gerbe, and let $\mathcal{Y}\to Y$ be a $G_Y$-gerbe. As the (abstract) groups $G_X$ and $G_Y$ are finite, the stacks $\mathcal{X}$ and $\mathcal{Y}$ are finite type separated DM stacks. Since $\mathcal{X}$ and $\mathcal{Y}$ are isomorphic, the coarse space of $\mathcal{X}$ is isomorphic to the coarse space of $\mathcal{Y}$. We are now done by the following lemma.

Lemma 1. The coarse space of $\mathcal{X}$ is $X$ and the coarse space of $\mathcal{Y}$ is $Y$.

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  • $\begingroup$ Can you maybe add some details to flesh out this answer? $\endgroup$
    – David White
    Jan 23, 2018 at 21:34
  • $\begingroup$ I only pointed it out because your answer wound up in the "low quality posts" review queue for being short. Also, you should probably make this CW, since that's the usual practice when turning someone else's answer-in-comments into an answer. $\endgroup$
    – David White
    Jan 23, 2018 at 22:41

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