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?
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.