3 latexification (answer orginally from MO early pre-latex days)

The classical (pre Deligne-Mumford) approach is to map Mg $\mathcal{M}_g$ into Ag $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that Ag $\mathcal{A}_g$ is quasi-projective is to define it as the Siegel upper half space Hg $\mathcal{H}_g$ (g $g$ by g $g$ complex matrices with positive define imaginary part), modulu SP(2g,Z); $\mathrm{SP}(2g,\mathbb{Z})$; We define the level cover Ag(m)->Ag, $\mathcal{A}_g(m)\to\mathcal{A}_g$, which is the moduli of Ag $\mathcal{A}_g$ plus torsion points, which is the quotient of Hg $\mathcal{H}_g$ by Gamma(m) $\Gamma(m)$ (the matrixes in SP(2g,Z) $\mathrm{SP}(2g,\mathbb{Z})$ which are trivial modulo n), $n$), and send Ag(m) $\mathcal{A}_g(m)$ to some projective space using polynomials in the theta constants.

Reference (for both Torreli and the embedding of Ag $\mathcal{A}_g$ 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 $H_1 \cap H_1 \cap cap$ canonical-curve is not 0).$0$).

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

2 added remark on quasi-projectiveness of Ag

The classical (pre Deligne-Mumford) approach is to map Mg into Ag using the Torreli map, and observe . Whereas the classical way to see that A_g itself Ag is a quasi-projective varietyis to define it as the Siegel upper half space Hg (g by g complex matrices with positive define imaginary part), modulu SP(2g,Z); We define the level cover Ag(m)->Ag, which is the moduli of Ag plus torsion points, which is the quotient of Hg by Gamma(m) (the matrixes in SP(2g,Z) which are trivial modulo n), and send Ag(m) to some projective space using polynomials in the theta constants.

Reference (for both Torreli and the embedding of Ag 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.

1

The classical (pre Deligne-Mumford) approach is to map Mg into Ag using the Torreli map, and observe that A_g itself is a quasi-projective variety.

Reference: 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.