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This question is inspired by What is an example of a function on M_g? . Consider Mg, the moduli space of genus g curves, NOT compactified. When g is 3 or greater, this is not affine. Does anyone know a good description of the ring of global functions on Mg, or of the spectrum of this ring?

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up vote 9 down vote accepted

By exercise 2.10 of Moduli of Curves by Harris and Morrison there exists a complete curve through any two points of M_g (this follows from the fact that the `boundary of the Hodge theory' compactification of M_g has codimension at least 2 and that M_g is quasi-projective). It follows the only global functions on M_g are constant.

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Doesn't M_g have codimension 2 in the Satake compactification for g > 2?

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