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.



Moduli spaces of complex curves as algebraic varieties
The classical (pre Deligne-Mumford) 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 quasi-projective 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 $H\_1 \cap H\_1
 \cap$ canonical-curve is not $0$).


He also says these are the only "coordinate oriented" methods he knows of.


