Functions on the moduli space of projective structures on complex curves

TL;DR

This paper studies the algebraic functions on the moduli space of projective structures on complex curves, showing that for genus g ≥ 3, the space admits no nonconstant algebraic functions, unlike the genus 2 case.

Contribution

It proves that the moduli space of projective structures on genus g ≥ 3 curves has no nonconstant algebraic functions, revealing a fundamental difference from the genus 2 case.

Findings

For g ≥ 3, the moduli space ${\\mathcal{P}}_g$ admits no nonconstant algebraic functions.

The case g=2 is different, with ${\mathcal{P}}_2$ being an affine variety.

The result highlights a contrast in the algebraic structure of these moduli spaces based on genus.

Abstract

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.