$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties 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 ???
 A: 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. 
