Applications of Faber's conjecture - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:15:10Z http://mathoverflow.net/feeds/question/21207 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21207/applications-of-fabers-conjecture Applications of Faber's conjecture Craig Westerland 2010-04-13T14:09:49Z 2010-04-13T17:20:47Z <p>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 $R^{g-2-k}$.</p> <p>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:</p> <ol> <li><p>Why might Faber have conjectured this to be the case? What is it about $R^*$ that suggests that it might satisfy Poincare duality?</p></li> <li><p>If true, what sort of applications does this have (to our understanding of $\mathcal{M}_g$, for instance)?</p></li> </ol> http://mathoverflow.net/questions/21207/applications-of-fabers-conjecture/21222#21222 Answer by VA for Applications of Faber's conjecture VA 2010-04-13T16:42:31Z 2010-04-13T17:20:47Z <ol> <li><p>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.</p></li> <li><p>That's not so clear. But that's a very mysterious property. The search for the meaning is on.</p></li> </ol>