Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves

TL;DR

We developed a computer program to compute the weight-graded rational cohomology of moduli spaces of pointed smooth curves, including boundary stratifications and symmetric group actions, for specific genus cases.

Contribution

The paper introduces a novel computational implementation of Deligne's spectral sequences to analyze tautological cohomology of moduli spaces.

Findings

Computed weight-graded cohomology for genus five moduli spaces.

Analyzed cohomology for genus three with three marked points.

Determined the action of symmetric groups on cohomology groups.

Abstract

We have written a computer program that implements Deligne's pullback and pushforward weight spectral sequences to compute the weight graded pieces of the rational cohomology of moduli spaces of pointed smooth curves (as well as curves of compact type and curves with rational tails) in cases where the cohomology groups appearing in the boundary stratification of the Deligne-Mumford compactification are generated by tautological classes (and when Pixton's relations are all relations). The weight graded pieces are computed together with the induced action of the symmetric group permuting the points on the curves. Using the computer program we have determined this information in the case of genus five as well as in the case of genus three with three marked points.