Cohomology of moduli spaces of stable curves
type: Item
instance of: scholarly article
title: Cohomology of moduli spaces of stable curves
zbMATH Open document ID: 0902.14017
author: Maurizio Cornalba
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/225820
full work available at URL: Cornalba.MAN.html
review text: \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).
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 1184282
zbMATH Keywords: moduli space of stable curves
zbMATH Keywords: rational cohomology ring
zbMATH Keywords: natural classes
zbMATH Keywords: \(n\)-pointed curves
zbMATH Keywords: tautological ring
reviewed by: Martin J. Pikaart
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1184282