Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford and Fogarty's book (Geometric Invariant Theory) is indicated as a reference for this statement. However, it is an advanced book for me. I cannot identify where this is proven in the book of Mumford and Fogarty. Can anyone help me locate me ???
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$\begingroup$ I think the answer might depend on why you want to know. (e.g. someone might instead recommend a later exposition.) $\endgroup$– Will SawinCommented May 9, 2020 at 0:38
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$\begingroup$ Can you recommend a reference, where it contains the proof that $\mathcal{M}_g$ and $\mathcal{A}_g$ are quasi-projective varieties, it would be good :) $\endgroup$– ManoelCommented May 9, 2020 at 1:25
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9$\begingroup$ In the 3rd edition of GIT, this is Theorem 7.10 for $\mathscr{A}_g$ and Corollary 7.14 for $\mathscr{M}_g$. The question would be more appropriate on MSE. $\endgroup$– abxCommented May 9, 2020 at 3:33
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The GIT proof gives very nice compactifications of these spaces (and is the "right" way to do this), but they were known to be quasiprojective varieties long before GIT was developed.
The classical proofs depend on properties of theta functions. For $\mathcal{A}_g$, it should be attributed to some combination of Satake and Baily, and the appropriate references are
Satake, Ichiro On the compactification of the Siegel space. J. Indian Math. Soc. (N.S.) 20 (1956), 259–281.
and
Baily, Walter L., Jr. Satake's compactification of Vn. Amer. J. Math. 80 (1958), 348–364.
A textbook reference for this is
J. Igusa, Theta functions, Springer, New York, 1972.
For $\mathcal{M}_g$, the first person to show that it was a quasiprojective variety was Baily. In fact, what he did was show that the Schottky locus in $\mathcal{A}_g$ is an open dense subset of its closure in the Satake compactification. The reference is
Baily, Walter L., Jr. On the moduli of Jacobian varieties. Ann. of Math. (2) 71 (1960), 303–314.
answered May 9, 2020 at 4:25
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1$\begingroup$ Do any of these references work in characteristic p, or was the proof in GIT the first proof in general? $\endgroup$ Commented May 9, 2020 at 4:33
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1$\begingroup$ @Anonymous: These proofs only work over $\mathbb{C}$. I think the GIT proof was the first one that worked over any other fields. $\endgroup$ Commented May 9, 2020 at 4:35
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4$\begingroup$ Note that it is not at all clear that the Schottky locus is the same as $\mathcal{M}_g$ (the problem is with the hyperelliptic locus). This was proved much later by Oort and Steenbrink. $\endgroup$– abxCommented May 9, 2020 at 6:06
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1$\begingroup$ @abx: That's a good point, though if you just care about being quasiprojective it is not a huge deal since the map from $\mathcal{M}_g$ to the Schottky locus is bijective, and thus certainly quasi-finite. $\endgroup$ Commented May 9, 2020 at 18:51