Birational aspects of the geometry of M_g

TL;DR

This paper explores the birational geometry of the moduli space of curves, providing new proofs and streamlined accounts of key results on its Kodaira dimension, unirationality, and general type status.

Contribution

It offers simplified proofs of the Harris-Mumford theorem, a self-contained account of Verra's work on unirationality, and discusses the general type of M_g for genus 22.

Findings

Proof of Harris-Mumford theorem using tautological Koszul bundles

Streamlined account of Verra's unirationality results

Proof that M_g is of general type for genus 22

Abstract

We discuss topics on the geometry of the moduli space of curves. We present a short proof of the Harris-Mumford theorem on the Kodaira dimension of the moduli space which replaces the computations on the stack of admissible covers by a simple study of tautological Koszul bundles on M_g. We also present a streamlined self-contained account of Verra's recent work on the unirationality of M_g. Finally, we discuss a proof that the moduli space of curves of genus 22 is of general type. Written for Surveys in Differential Geometry.