This question is inspired by [What is an example of a function on M\_g?](https://mathoverflow.net/questions/453/what-is-an-example-of-a-function-on-mg) . 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?



What is the affinization of M_g?
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.


