Carel Faber & Dan Petersen: The Gorenstein conjecture fails for the tautological ring of stable n-pointed genus two curves

Carel Faber:  I will recall the formulation of the conjectures and summarize the results obtained on them.

Dan Petersen:  I discuss recent joint work with Orsola Tommasi, in which we show that the  tautological ring of the moduli space of stable n-pointed genus two curves is not Gorenstein in general. In fact, as soon as there are non-tautological cohomology classes in even degree on this space, the Gorenstein conjecture fails. Graber and Pandharipande constructed such classes on \bar M_{2,20}; we give an alternative construction. We expect that both constructions produce the same classes, and that \bar M_{2,20} is actually the first case where there is non-tautological cohomology in even degree.