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

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.