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 $g\, \geq\, 3$, 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.