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Faber's perfect pairing conjecture states that the tautological ring $R^*$ of the moduli space $\mathcal{M}_g$ of curves of genus $g$ behaves as if it were the rational cohomology of a closed, oriented manifold of dimension $g-2$. Specifically, $R^{g-2}$ is rank one, and multiplication into this degree gives a perfect pairing between $R^k$ and $R^{g-2-k}$.

My understanding is that it is known (through work of Looijenga, Faber, and Pandharipande) that $R^{g-2} = \mathbb{Q}$, but the perfect pairing part hasn't been proven (though it has been verified in low genus cases). I'd like to know:

  1. Why might Faber have conjectured this to be the case? What is it about $R^*$ that suggests that it might satisfy Poincare duality?

  2. If true, what sort of applications does this have (to our understanding of $\mathcal{M}_g$, for instance)?

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I could be mis-remembering, but I thought that the conjecture says that the tautological ring looks like the cohomology ring of a projective variety, not just a closed oriented manifold. This would give quite a bit more structure. – Jeffrey Giansiracusa Apr 13 '10 at 16:11
up vote 6 down vote accepted
  1. Numerical evidence, from computing the cases $g=2,3,\dots$, eventually $g\le 15$, and seeing the symmetry in the numbers $\dim R_g^n$. I recall Carel saying he made the conjecture when $g$ was still pretty low, maybe 6. For any $g$, there is an algorithm computing $\dim R^n_g$ in finite time, that Faber came up with.

  2. That's not so clear. But that's a very mysterious property. The search for the meaning is on.

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Wow; that's surprising. Do you know: has anyone constructed submanifolds of $M_g$ whose cohomology realizes $R^*$ in these low genus cases? – Craig Westerland Apr 14 '10 at 0:44
To the best of my knowledge, no such submanifolds are known. – VA. Apr 14 '10 at 1:14

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