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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

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up vote 2 down vote accepted

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?

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Thanks a lot! Harris Morrison says indeed that if you take Satake compaction $\tilde M_g$ of $M_g$ then first of all it is projective and second $\tilde M_g\setminus M_g$ has codimension $2$ provided $g\ge 3$ (a bit surpising :)) . So there are plenty of curves on $\tilde M_g$ that miss $\tilde M_g\setminus M_g$. –  aglearner Nov 6 '12 at 14:38
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