Rational cohomology of the moduli space of genus 4 curves

TL;DR

This paper computes the rational cohomology of the moduli space of smooth genus 4 curves by constructing a stratification and analyzing geometric quotients, revealing its Poincare polynomial.

Contribution

It provides the first explicit calculation of the Poincare polynomial for genus 4 moduli space using stratification and quotient techniques.

Findings

Poincare polynomial is 1+t^2+t^4+t^5

Stratification of the moduli space into geometric quotients

Method applicable to similar moduli space computations

Abstract

We prove that the Poincare' polynomial of the moduli space of smooth genus 4 curves is 1+t^2+t^4+t^5. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.