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)?



Applications of Faber's conjecture
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.


