What is the calculation that shows that the Hilbert or Quot functors could be represented by schemes (under various noetherian, (quasi) projectivity hypotheses), and do not require extending to the category stacks? For example, going into the world of stacks seems to be necessary because there can be isotrivial families of elliptic curves. I'm guessing the data that the hilbert or quot functors rigidify things so there are no isotrivial families. This should be some short (known) deformation theory calculation, and my question is what is it. I'm not asking for the full proof of representability by schemes (I know where to find that), I'm asking why the hilbert and quot functors take their values in Sets as opposed to groupoids.
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$\begingroup$ The main reason is that the objects they classify do not have automorphisms. $\endgroup$– SashaCommented Mar 8, 2016 at 16:41
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$\begingroup$ @sasha yes but I'm asking why the don't have automorphisms? $\endgroup$– usr0192Commented Mar 8, 2016 at 16:46
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2$\begingroup$ Because they classify not just objects, but quotient objects. The morphism from $O_X$ (in case of Hilbert schemes) does the rigidification. $\endgroup$– SashaCommented Mar 8, 2016 at 16:48
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$\begingroup$ Are you asking about the Hilbert and Quot functors from my article with Olsson? Those functors manifestly take values in sets, rather than in groupoids. So there is no question that, when they happen to be representable, they are representable by algebraic spaces rather than general algebraic stacks. $\endgroup$– Jason StarrCommented Mar 8, 2016 at 18:44
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$\begingroup$ @JasonStarr Yes, those functors. I mean, what confusing me was a paper by Hall and Rydh where they talk about the Hilbert stack HIlb_{X/S}, where $X, S$ are stacks - but there I guess since stacks form a 2-category, morphisms (such as closed immersions in the case of HIlbert schemes) can have automorphisms, so that's why they need to consider the Hilbert stack. $\endgroup$– usr0192Commented Mar 8, 2016 at 23:29
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