Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. (English) Zbl 0886.14007

Arbarello, Enrico; Cornalba, Maurizio

J. Algebr. Geom. 5, No. 4, 705-749 (1996).

Let \(\overline {\mathcal M}_{g,n}\) denote the moduli space of \(n\)-pointed stable curves of genus \(g\). There are combinatorical cycles on \(\overline {\mathcal M}_{g,n}\) which are defined using Strebel’s theory of quadratic differentials. On the other hand there are algebro-geometric cohomology classes of \(\overline {\mathcal M}_{g,n}\), the so-called Mumford-Morita-Miller classes, defined via the relative dualizing sheaf of the universal curve over \(\overline {\mathcal M}_{g,n}\). It was first conjectured by Witten that the classes of the combinatorical cycles can be expressed in terms of the Mumford-Morita-Miller classes. The intersection numbers of the combinatorical cycles are best organized as coefficients of an infinite series \(F\) in infinitely many \(s\)- and \(t\)-variables. P. Di Francesco, C. Itzykon, and J.-B. Zuber proved in Commun. Math. Phys. 151, No. 1, 193-219 (1993; Zbl 0831.14010) that the \(s\)-derivatives of \(F\) are linear combinations of the \(t\)-derivatives of \(F\) evaluated at the same point. The authors conjecture that this result interpreted geometrically should provide the link between the combinatorical and algebro-geometric classes on \(\overline {\mathcal M}_{g,n}\). The main result of the paper is the proof of this conjecture in codimension 1. Moreover in some cases the authors give an explicit expression for the Di Francesco-Itzykson-Zuber-correspondence. In these cases the correspondence gives an explicit expression for the intersection numbers of the combinatorial cycles in terms of the algebro-geometric classes.

Reviewer: H.Lange (Erlangen)

MSC:

14H10	Families, moduli of curves (algebraic)
14H15	Families, moduli of curves (analytic)

Keywords:

Mumford classes; moduli space of \(n\)-pointed stable curves; combinatorical cycles; quadratic differentials