Using a minimum of technical vocabulary, give a summary of why it is that the moduli space of genus g complex curves with n marked points has a natural compactification that is isomorphic (as a complex orbifold) to a projective algebraic variety.

$\begingroup$ Well, actually what you write is not precisely true. In general you don't get a complex manifold, just because the thing you get is not smooth. But it is a smooth orbifold. $\endgroup$– Kevin H. LinOct 20, 2009 at 22:26

$\begingroup$ I changed 'manifold' to 'orbifold' (you still have a complex structure, right?) $\endgroup$– Jonah SinickOct 21, 2009 at 16:17
4 Answers
The classical (pre DeligneMumford) approach is to map $\mathcal{M}_g$ into $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}_g$ is quasiprojective is to define it as the Siegel upper half space $\mathcal{H}_g$ ($g$ by $g$ complex matrices with positive define imaginary part), modulu $\mathrm{SP}(2g,\mathbb{Z})$; We define the level cover $\mathcal{A}_g(m)\to\mathcal{A}_g$, which is the moduli of $\mathcal{A}_g$ plus torsion points, which is the quotient of $\mathcal{H}_g$ by $\Gamma(m)$ (the matrixes in $\mathrm{SP}(2g,\mathbb{Z})$ which are trivial modulo $n$), and send $\mathcal{A}_g(m)$ to some projective space using polynomials in the theta constants.
Reference (for both Torreli and the embedding of $\mathcal{A}_g$ above) : Mumfords curves and their Jacobians (now bundled together with the red book in LNM 1358) Lecture IV.
In Lecture II (same place) Mumford sketches two more "coordinate oriented" methods
 Tracking the Weierstrass point of curves.
 Tracking invariants of the Chow form of the canonical curve (the Chow form is the equation for pairs of hyperplanes such that $H_1 \cap H_1 \cap$ canonicalcurve is not $0$).
He also says these are the only "coordinate oriented" methods he knows of.

$\begingroup$ David  Thanks for the remark and especially for the reference to the Mumford book. If it's not too hard to do so, could you expand on why it suffices exhibiting Mg as a (suitable kind of) subset of a quasiprojective variety and why Ag is a quasiprojective variety? $\endgroup$ Oct 20, 2009 at 22:20

1$\begingroup$ As for why being a locally closed subset of a quasiprojective variety, you can then embed the quasiprojective into projective space, and then the locally closed subset will be locally closed in P^n, and so its closure is projective. As for why A_g is quasiprojective...can't help. $\endgroup$ Oct 20, 2009 at 22:26

$\begingroup$ @Johan: Charles is of course correct about your first question, and I edited my answer to answer the second; sorry about the delay  I'm on the other side of the globe. $\endgroup$ Oct 21, 2009 at 6:02

I'm not sure how useful this will be, but I can say a couple of words about why A_g is quasiprojective (I am very much not an expert, so real experts are welcome to correct me). This will be an outline of what happens in the proof, but without going into the details I can't explain why this works.
There are probably fancier ways of doing this, but the most classical is via the theta constants. The best complete reference for this is Igusa's book "Theta Functions", but a large part of the story can also be found in Mumford's "Tata Lectures on Theta I", which is easier to read. In particular, the 1dimensional case is worked out there very explicitly, and if you want to understand the details I would try to understand this case first. By the way, in the 1dimensional case we have A_{g} = M_{g} since every elliptic curve is isomorphic to its Jacobian.
The basic idea is that theta functions provide a natural set of coordinates on A_g. These coordinates give an embedding into projective space, and the compactification is the closure of this embedding.
A point of A_g is a complex tori T that happens to be a projective variety (together with a principal polarization, but don't worry about that now). A theta function on T is a holomorphic function on C^g (the universal cover of T) that satisfies a simple transformation law (it is really a section of a certain bundle on T lifted to the universal cover C^g). The set of all theta functions on T spans a finitedimensional vector space, and you can use them to embed T into projective space.
Our goal, though, is to embed A_g into projective space. The universal cover of A_g is the Siegel upper half plane H_g, which is a nice complex domain. In fact, it is the set of all skewsymmetric gxg complex matrices whose imaginary part is positive definite (for g=1, this is exactly the classical upper half plane). One now looks at the set of all holomorphic functions F(.,.) on H_g x C^g such that for all T in H_g, the function F(T,.) is a theta function on T. This again spans a finitedimensional vector space. Every PPAV has a distinguished point, namely 0. The "theta constants" or "theta nulls" are the functions F(.,0) on H_g. These satisfy a (somewhat complicated) transformation law on H_g coming from the fact that they are sections of a certain bundle on A_g. These thetanulls give you an embedding into projective space, and thus a compactification.

$\begingroup$ I think the only non accurate part here is that you are using torsion points (and in fact, as far as I recall, you need more then 2torsion), so what you get is a representation of a finite cover of Ag, and not Ag itself $\endgroup$ Oct 21, 2009 at 16:20

$\begingroup$ That's right  you are really working with the A_g(L) for L at least 3 so that you are working with a smooth space rather than an orbifold (here A_g(L) is the moduli space of PPAV's with level L structures). $\endgroup$ Oct 21, 2009 at 17:23
Technically I recall from the Tata lectures, that it is Ag(4,8) that embeds via theta nulls. Another nice reference is Freitag's book on Siegel modular functions.
The moduli space $\overline{M}_{g,n}$ has a structure of orbifold or in algbraic terms of DeligneMumford algebraic stack.
Since the deformations of $n$pointed genus $g$ DeligneMumford stable curves (at most nodal singularities and finite automorphism group) are unobstructed the stack is smooth of dimension $3g3+n$.
The corresponding coarse moduli space $\overline{M}_{g,n}$ is a projective variety with quotient singularities at the places where the automorphisms groups of the curves jump.
To understand why it is projective one can consider the usual GIT construction. One can embed a curve of genus $g\geq 2$ in $\mathbb{P}^{N}$ with the sections of the $3$canonical system. The action of $SL(N)$ on the Hilbert scheme $H$ of such curves can be linearized. Then one construct $\overline{M}_{g}$ as $H/SL(N)$. Now the projectivity follows from the projectivity of $H$ and standard theorems of GIT.