Family over the coarse moduli space of curves - MathOverflowmost recent 30 from mathoverflow.net2022-01-27T02:48:26Zhttps://mathoverflow.net/feeds/question/371070https://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/3710703Family over the coarse moduli space of curvesFabian Ruoffhttps://mathoverflow.net/users/1647822020-09-07T08:21:14Z2021-04-09T12:58:43Z
<p>Let <span class="math-container">$k$</span> be an algebraically closed field. As the coarse moduli space of curves <span class="math-container">$M_g$</span> of genus <span class="math-container">$g$</span> over <span class="math-container">$k$</span> is not a fine moduli space, it does not have a universal family. But I am wondering if it has a family (proper and flat) such that the fiber over every point <span class="math-container">$[C]$</span> of <span class="math-container">$M_g$</span> is isomorphic to the curve <span class="math-container">$C$</span>.</p>
<p>As a disclaimer: I am not that familiar with the language of stacks. As far as I understand the situation in this context, the stack <span class="math-container">$\mathcal{M}_g$</span> has an universal family <span class="math-container">$\mathcal{C}_g$</span>. The corresponding coarse moduli space of <span class="math-container">$\mathcal{C}_g$</span> is <span class="math-container">$M_{g,1}$</span>, so the coarse moduli space of curves with one marked point. The morphism <span class="math-container">$\pi \colon M_{g,1} \to M_g$</span> on the level of quasiprojective varieties is just forgetting about the marked point. This family has the property that the fiber over a point <span class="math-container">$[C]$</span> is isomorphic to <span class="math-container">$C$</span>, at least if <span class="math-container">$C$</span> has no nontrivial automorphisms. In all other cases the fiber is isomorphic to <span class="math-container">$C/\operatorname{Aut}(C)$</span>. Is it possible to get something better than that?</p>
https://mathoverflow.net/questions/371070/-/389732#3897321Answer by Fabian Ruoff for Family over the coarse moduli space of curvesFabian Ruoffhttps://mathoverflow.net/users/1647822021-04-09T12:58:43Z2021-04-09T12:58:43Z<p>To close up loose ends and for everyone finding this questions: Such a family does not exist in general. An argument for elliptic curves can be found in Robin Hartshorne Deformation Theory in Remark 26.3.1.</p>