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.

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

We prove that all monomials of κ-classes and ψ-classes are independent in for all . We also give a simple argument for in Rl(ℳg) for .