I think that for $n \geq 3$, the Deligne-Mumford moduli stack $\mathcal{M}_{0,n}$ is a scheme. Is it more generally true that for every $g$, the Deligne-Mumford moduli stack $\mathcal{M}_{g,n}$ is a scheme for $n \gg 0$? (My intuition is that the presence of many disjoint marked points "kills" automorphisms.)
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1$\begingroup$ This is covered in mathoverflow.net/questions/11253/… . See in particular JSE's answer. $\endgroup$– S. Carnahan ♦Commented Mar 21, 2015 at 17:06
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$\begingroup$ In this thread, Emerton only addresses this question for elliptic curves, and JSE only the question about automorphism groups (not the representability by a scheme). $\endgroup$– user19475Commented Mar 21, 2015 at 18:12
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1$\begingroup$ Bjorn's comment under JSE's answer addresses the representability question. The absence of automorphisms automatically yields an algebraic space, but the coarse moduli space is known to be a quasi-projective variety, so you get a scheme. $\endgroup$– S. Carnahan ♦Commented Mar 21, 2015 at 19:23
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1$\begingroup$ @TimoKeller To answer your question in the comments of the other question, see Corollary 8.1.1 in the book Champs Algebriques by Laumon and Moret-Bailly: Any DM stack with trivial stabilizers is an alg space. To see that an alg space is a scheme you can apply Knudson's criterion sometimes and use properties of the coarse moduli space as S. Carnahan mentions. (For Knudson's criterion see Cor. II.6.16 of his book on Alg Spaces.) $\endgroup$– Ariyan JavanpeykarCommented Mar 22, 2015 at 22:45
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