Dan Petersen: More counterexamples to the Gorenstein conjecture

Faber and Pandharipande made a "trinity" of conjectures regarding the tautological rings of moduli spaces of curves. Specifically, they conjectured that we have 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. Last year I proved with Orsola Tommasi that in the stable case this conjecture is false for g=2 and some n. I will explain that there is now also a counterexample in the compact type case: the tautological ring does not have Poincaré duality when g=2 and n=8.