The rationality of the moduli space of curves of genus 3 after P. Katsylo

TL;DR

This paper surveys P. Katsylo's proof that the moduli space of genus 3 curves is rational, aiming to clarify the geometric ideas and structural features involved in the proof.

Contribution

It provides a clearer, more accessible explanation of Katsylo's original proof of the rationality of the genus 3 moduli space.

Findings

The moduli space of genus 3 curves is rational.

Key geometric structures underpin the proof.

The proof's core ideas are made more transparent.

Abstract

This article is a survey of P. Katsylo's proof that the moduli space of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features.