Cohomology of moduli spaces of stable curves (Q1126725)

\textit{E. Arbarello} and \textit{M. Cornalba} [J. Algebr. Geom. 5, No. 4, 705-749 (1996; Zbl 0886.14007)] defined so-called natural or tautological classes in the rational cohomology ring \(H^*(\overline M_{g,n},\mathbb Q)\) of the moduli space of stable complex \(n\)-pointed curves of genus \(g\). These classes generate the tautological ring. The author first discusses recent progress about this ring in low degree. He explains in detail the Lefschetz- and Hodge-type arguments used to compute \(H^{k}(\overline M_{g,n},\mathbb Q)\) for \(k \leq 5\). Secondly he studies the low genus case, concentrating on the common ideas underlying the computation of a generating function for the Euler characteristics \(\chi(\overline M_{g,n})\) for \(g=1\) and \(g=2\). For a more comprehensive overview of the cohomology of the moduli space of curves, see \textit{R. Hain} and \textit{E. Looijenga} in: Algebraic geometry. Proc. Summer Res. Inst., Santa Cruz 1995, Proc. Symp. Pure Math. 62, part 2, 97-142 (1997).