Faber and Pandharipande formulated a ``trinity'' of conjectures
regarding the tautological rings of moduli spaces of
curves. Specifically, they conjectured that there is Poincaré
duality in the tautological ring of the space of \(n\)-pointed genus
\(g\) curves that are either (i) stable, or (ii) of compact type, or
(iii) with rational tails. I will explain that there are now two
known counterexamples to this conjecture: in the stable case, it
fails when \(g=2\) and \(n \geq 20\) (this is due to joint work with
Orsola Tommasi), and in the compact type case, it fails when \(g=2\)
and \(n \geq 8\).
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