Let $M\_3$ be the moduli space of genus three curves over $\mathbb C$. 
Are there non-constant regular functions of this space? What about complex analytic functions?


This question is prompted by the following one :
[Does the moduli space of genus three curves contain a complete genus two curve](https://mathoverflow.net/questions/111556/does-the-moduli-space-of-genus-three-curves-contain-a-complete-genus-two-curve)



Regular (or complex analytic) functions on M_3
Here's an older MO post addressing this question (which appears to apply to M\_3, though I don't have Harris-Morrison handy):
[What is the affinization of M\_g?](https://mathoverflow.net/questions/559/what-is-the-affinization-of-mg/)


