Functions on the moduli space of projective structures on complex curves

We investigate the moduli space ${\mathcal{P}}_g$ of smooth complex projective curves of genus $g$ equipped with a projective structure. When $g\ge 3$, it is shown that this moduli space ${\mathcal{P}}_g$ does not admit any nonconstant algebraic function. This is in contrast with the case of ${\mathcal{P}}_2$ which is known to be an affine variety.