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It feels bad talking about a space without knowing a single function on it, hah?

So what is a function on the moduli space of curves, from the geometric point of view?

From the functorial point of view, it should be invariants of family of curves, which is natural w.r.t. pullback of families. The j-invariant maybe one for M_1. But does anybody have a concrete example for higher genus?

I do have two guesses of sources of functions:

  1. There are some geometrically defined divisors, maybe take one, and then pick two sections of the line bundle it defines, and take their quotient?

  2. Maybe there are some "natural" differential form, whose integral over the whole curve is a function on M_g?

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i think there are no non-constant holomorphic function on them.

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I think you are right. I interpreted the question to mean maps to P^1. – David Zureick-Brown Oct 15 '09 at 5:18
Really? Are you sure you aren't thinking of a compactification. – Anton Geraschenko Oct 15 '09 at 5:29

The function field of M_2, at least, is completely explicitly understood; it is free of transcendence degree 3 (which follows from rationality, of course) and an explicit set of generators is given by the Igusa invariants of genus-2 curves, which are the analogue of "j" in this setting.

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You can go much higher than 5, I think that the highest genus for which M_g known to be rational is somewhere around 14 and the lowest for which it is known not to be is somewhere around 23. See "Moduli of Curves", the beginning of Section 6.F, by Harris and Morrison, for very nice explicit descriptions of M_g for low g.

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If you only care about CONTINUOUS functions, they you can manufacture interesting ones using hyperbolic geometry. The key is that every compact Riemann surface of genus at least 2 has a canonical hyperbolic metric. There is thus a continuous function L which takes a Riemann surface S to the length of the shortest closed geodesic on S with its natural hyperbolic metric. A beautiful theorem of Mumford says that for all e>0, the subset of moduli space where L is at least e is compact (this region is known as the "thick" part of moduli space). This can be viewed as an analogue for M_g of Mahler's compactness criteria.

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For g=2, you can make functions using the fact that genus 2 curves are hyperelliptic. There is a finite map from an open subset of (P^1)^3 to M_2 given by taking the 2-fold cover of P^1 branched at 0,1, infinity, and 3 additional points. Then you can integrate any function on this open subset along the map.

For small g (known for g less than 5, I think), the coarse space of M_g is rational, so if you only care about meromorphic functions, you're all set.

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The period mapping lets you embed $M_g$ in the quotient of the Siegel upper half-space by the integral symplectic group; so any function on this locally symmetric space restricts to a function on $M_g$.

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It's an interesting question, but it's not obvious knowing a function will help you a lot. E.g. consider a similar question: it feels bad talking about elliptic curve without explicitly writing its equation.

For (2), perhaps, take g natural forms, then g natural cycles and take determinant of their products. I don't remember if there are g natural forms. I think this gives you a real number up to (p/q)^2, called period, but still not a function.

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Any odd theta characteristic is realized in the canonical system as a hyperplane (dualizing system if you want to work on the boundary too). So, for each curve you get a set of N=(2^g-1)(2^(g-1)) points in P^(g-1), possibly with multiplicities but who cares. Moreover, by a paper of Caporaso and Sernesi the map we just constructed (from M_g to Sym^N P^{g-1}/PGL_g) is genrically 1-1. All you have to do now is write down all the PGL_g times S_n symmetric functions, and your are done (by the way this is the way Igusa comupted the invariant theory of M_2).

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