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1. There are a lot of works concerning the intersection theory on the moduli spaces of curves $\mathcal M_{g,n}$ (and their Deligne-Mumford compactifications $\overline{\mathcal M}_g$), for $n>0$.

2. An interesting part of the theory consists in working with combinatorial models $\mathcal M_{g,n}^{\rm comb}$ constructed from the classical $\mathcal M_{g,n}$'s by mean of fatgraphs (using the conformal Harer-Mumford-Strebel's approach, or Penner's decorated Teichmüller theory).


What about the case when $n=0$?

More precisely:

Question 1: concerning intersection theory on the $\mathcal M_{g}$'s (and the $\overline{\mathcal M}_g$'s): what is known? What are the main open questions?

Question 2: as in the case when $n>0$, are there combinatorial models $\mathcal M_{g}^{\rm comb}$ of the $\mathcal M_{g}$'s?

Thanks in advance for any help.

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  • $\begingroup$ What kind of intersection theory are you interested in? Are you thinking about cohomology, and if so, do you want ordinary or compact support? Are you thinking about the "stable range" of degrees for the given genus, i.e., the Miller-Morita-Mumford conjecture, proved by Madsen-Weiss? Are you interested in the "tautological ring", e.g., the Faber conjecture? Are you interested in the part of the cohomology ring that is non-algebraic? $\endgroup$ – Jason Starr Dec 3 '13 at 11:58
  • $\begingroup$ @Jason Starr: this is the point! I'm not at all aware of anything concerning any kind of intersection theory of the moduli spaces $\mathcal M_g$'s..! A good answer to Question 1 would be a kind of survey covering the (or some of the) conjectures/theorems you mention. But maybe that MO is not the best place for this. Can you indicate some references? Thanks. $\endgroup$ – Lucien from IHP Dec 3 '13 at 16:08
  • $\begingroup$ We still have forgetful maps $\overline{\mathcal{M}}_{g,n}\to \overline{\mathcal{M}}_g$, using which cohomology classes can be pushed into M_g. Is that what you want? $\endgroup$ – Mohammad Farajzadeh-Tehrani Dec 3 '13 at 21:56
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l will try to answer the first question in a way so that you can guess the answer of the second. There are reasons why $\bar{M}_{g}$ is more preferred since calculations with compact spaces is easier. So l will focus on the (Deligne-Mumford) compactified moduli of curves and this involves stable curves ie curves with finite automorphism group. Finiteness condition of the automorphism group is equivalent to the condition that a curve with $n$ marked points of genus $g$ must satisfy $2-2g-n<0$ or for components $2-2g_{i}-n_{i}<0$. Let us look at the moduli of curves with marked points at first $\bar{M}_{g,n}$. If you want to do intersection theory on this you need to find out the cycles and the characteristic classes they represents. Intersection theory involves cohomology classes(because of the connection of Poincare dual classes with the cup product). If you consider the pairing $<,> : H^{*}(M,\mathbb{Z})\times H_{*}(M,\mathbb{Z}) \rightarrow \mathbb{Z} $ on a smooth oriented manifold $M$, it can be written as $<a,B>=\int_{B}a $ which is an integer. We want to do something similar with the moduli "space"(actually a stack) $\bar{M}_{g,n}$ which has the dimension $dim_{\mathbb{C}}=3g-3+n$. It has the fundamental class $[\bar{M}_{g,n}]\in H_{2(3g-3+n)}(\bar{M}_{g,n},\mathbb{Q}), $ since this is a stack or in the analytic setting a complex orbifold we should consider (co)homology coeficcients in the rational numbers. Then the (top) intersection numbers should also be rational numbers which are $<[\bar{M}_{g,n}],\alpha_{1} ...\alpha_{k} > = \int_{\bar{M}_{g,n}}\alpha_{1}...\alpha_{k} \in \mathbb{Q}$

Actually for moduli spaces working with the cohomology ring is quite cumbersome, instead we consider the tautological ring containing almost all the geometrically interesting classes.

For example the psi-classes $\psi_{i}= c_{1}(\mathbb{L}_{i}) \in H^{2}(\bar{M}_{g,n}, \mathbb{Q})$ where $\mathbb{L}_{i}$ is the $i^{th}$ line bundle (relative dualising sheaf actually) and there are lambda classes which are chern classes of the hodge bundle.

The point l want to emphasize here is the relative dualising sheaf. If you want to do intersection theory on the moduli $\bar{M}_{g}$ with no marked points you still need to condsider a moduli space of curves with one marked point $\bar{M}_{g,1}$ because the universal curve $\bar{C}_{g}$ in this case is $\bar{M}_{g,1}$.

You can look at Mumfords paper "Towards an enumerative geometry of the moduli space of curves" part 2, where he defined the tautological classes for $\bar{M}_{g}$ !

He considers the relative dualizing sheaf $\omega_{\bar{C}_{g} / \bar{M}_{g}}$ for the map $\bar{C}_{g} \rightarrow \bar{M}_{g}$. It is what we call the forgetful morphism of the moduli space which forgets a marked point and stabilizes the curve if needed. In that sense studying intersection theory on $\bar{M}_{g,n}$ is a more generalized setting of which $\bar{M}_{g}$ is just a special case.

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