On the rational Picard group of the moduli space of curves

TL;DR

This paper discusses a potential algebraic proof of Harer's theorem on the rational Picard group of the moduli space of smooth complex curves, refining existing approaches involving Hurwitz spaces.

Contribution

It proposes a new algebraic approach to prove Harer's theorem, building on and refining the Diaz and Edidin method using Hurwitz spaces.

Findings

Refined approach to the Picard group of moduli space

Potential algebraic proof of Harer's theorem

Enhanced understanding of Hurwitz space parameterizations

Abstract

We speculate about an algebro-geometric proof of Harer's theorem on the rational Picard group of the moduli space of smooth complex curves. In particular, we refine the approach of Diaz and Edidin involving the Hurwitz space which parameterizes smooth covers of the projective line.