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Could anyone give some interesting motivations to understand the cohomology of $\mathcal{M}_g$?

What I know: I have read the various approaches to construct $\mathcal{M}_g$ via orbit spaces for group actions, period maps, and Teichmüller theory. I learned from literature that the cohomology of $\mathcal{M}_g$ is an important object to study, though I have not started reading some classical papers on the topic, such as Mumford's Towards an enumerative geometry of the moduli space of curves, Miller's The homology of the mapping class group, or Morita's Characteristic classes of surface bundles.

What aspects am I interested in: Some notes mentioned that the cohomology of moduli space is an important source of motives. Although I do not really understand what is a motive, any elaboration in this direction would be appreciated. Besides that, I am more interested in any concrete applications in algebraic geometry, differential geometry, or topology to motivate the study of cohomology of $\mathcal{M}_g$.

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  • $\begingroup$ It would really help if you outline your background, and tell us what you already know beyond a general phrase "has been a subject of interests to people from different fields for a long time". Putting more effort in a question increases chances for a meaningful answer... $\endgroup$
    – Vladimir Dotsenko
    Commented Jul 5, 2017 at 23:40
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    $\begingroup$ Thanks for the suggestions, @VladimirDotsenko. I have edited my questions following your guideline. $\endgroup$
    – Taisong Jing
    Commented Jul 7, 2017 at 0:18
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    $\begingroup$ In relation to Jim Bryan's very nice answer, I would mention the following slogan: "To compute a cycle class on a moduli space is to solve a universal enumerative problem." So for example, we know the classes of the Brill-Noether and Gieseker-Petri divisors due to work of Eisenbud, Harris, and Mumford; if we have a 1-parameter family of curves, we can compute the number of B-N or G-P special fibers in the family simply by computing the degrees of $\lambda$ and the boundary classes on this family and plugging these numbers into the divisor class. $\endgroup$
    – Tabes Bridges
    Commented Jul 9, 2017 at 3:24

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As the title to Mumford's famous paper "Toward an enumerative geometry..." suggests, knowing the cohomology / cycle theory of the moduli space of curves allows one to answer enumerative geometry questions for curves.

Here is an (very concrete) example that came up in real life for a student of mine. He had a family of genus 2 curves over $\mathbb{P}^2$ constructed by taking the family of lines in $\mathbb{P}^2$ and for each line taking the double cover of the line branched at the six points given by the intersection of the line with a fixed (generic) sextic curve (equivalently, this is the defining linear system on a generic genus 2 $K3$ surface). He needed to know how many of those genus two curves admit a degree $d$ map to a fixed elliptic curve $E$. This is a calculation in the cohomology ring of $\overline{M}_2$: we are looking for the number of points in the intersection of the two cycles in $\overline{M}_2$ given by the two families of curves: the 2-cycle given by the family over $\mathbb{P}^2$ and the 1-cycle given by the locus of genus 2 curves admitting a degree $d$ map to $E$ (since the dimension of $\overline{M}_2$ is 3, we expect these cycles to intersect in points). This intersection problem is dual to a cup product computation in the cohomology ring of $\overline{M}_2$. Using Mumford's complete description of this cohomology ring, one can determine how to express each of our cycles in terms of Mumford's generators and then compute the cup product. There are some minor issues to fuss with involving the orbifold structure and some general position issues, but the problem is essentially solved with the cohomology of the moduli space.

This was a rather specific example, but maybe you get the general idea: if we want to count the "number of curves satisfying this that and the other", we try to formulate the question as a problem in the cohomology ring of the moduli space.

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  • $\begingroup$ Naive question - How do you see that the space of genus 2 curves admitting a degree $d$ map to $E$ is 1-dimensional? (ie, why is it a 1-cycle)? $\endgroup$
    – Will Chen
    Commented Jul 7, 2017 at 5:07
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    $\begingroup$ @oxeimon: up to a finite choice, such a curve is determined by the branch divisor, which in this case consists of 2 points, up to automorphisms of $E$: this gives a one-parameter family. $\endgroup$
    – abx
    Commented Jul 7, 2017 at 8:20
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    $\begingroup$ Right, and the number of branched points is determined by the Riemann-Hurwitz formula which in this case turns out to be independent of the degree $d$ of the map $C\to E$. So the above enumerative problem is really a family of problems indexed by $d$. It turns out that $\sum_d n_d q^d$, the generating function for the solutions to that problem, is equal to a quasi-modular form! $\endgroup$
    – Jim Bryan
    Commented Jul 7, 2017 at 14:19
  • $\begingroup$ @JimBryan Interesting... What is your definition of a quasi-modular form? (the definition on wikipedia doesn't seem to make sense with what you said). Also, do you know the level of the form? (ie, the subgroup of SL(2,Z) that stabilizes it)? $\endgroup$
    – Will Chen
    Commented Jul 7, 2017 at 16:35
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    $\begingroup$ @oxeimon what I said should make more sense with the wikipedia page if you set $q=e^{2\pi i \tau}$. Then the generators of the ring of quasi-modular forms, namely the Eisenstein series $E_2$, $E_4$, and $E_6$ have $q$ expansions where the coefficients are integers. I can't remember the exact formulas (my student has them) but I seem to remember that it involved $E_2^2$ and so it should be weight 4. $\endgroup$
    – Jim Bryan
    Commented Jul 8, 2017 at 17:39

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