Tautological rings of spaces of pointed genus two curves of compact type

Abstract

We prove that the tautological ring of M ct 2,n , the moduli space of n-pointed genus two curves of compact type, does not have Poincaré duality for any n ≥ 8. This result is obtained via a more general study of the cohomology groups of M ct 2,n . We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of H k (M ct 2,n ) for any k and n considered both as Snrepresentation and as mixed Hodge structure/ℓ-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of M 2,n is tautological for n < 20, and that the tautological ring of M 2,n fails to have Poincaré duality for all n ≥ 20. This improves and simplifies results of the author and Orsola Tommasi.

A calculus for the moduli space of curves

“…While question Q2 is completely open, a negative answer would be surprising since many different mathematical approaches have failed to find relations outside of the Faber-Zagier span [18,40,70,72,82,87]. Moreover, the Gorenstein property for the algebra of tautological classes has been proven to fail for the moduli space M ct 2,8 of curves of compact type and the moduli space M 2,20 of stable curves in [74,78]. So there appears to be no compelling reason to believe the Gorenstein property holds for M 24 .…”

Tautological classes on the moduli space of hyperelliptic curves with rational tails

“…In [29] Petersen and Tommasi show the existence of a counterexample to the Gorenstein conjectures for M 2,n for some n ≤ 20. A more recent result [28] of Petersen shows that the tautological ring of M ct 2,n is not Gorenstein when n ≥ 8. These results suggest that the structure of the tautological rings of the larger compactifications become more complicated already in genus two.…”

Poincaré Duality of Wonderful Compactifications and Tautological Rings

“…We remark that it is likely that R • (M rt g,n ) does not have Poincaré duality in general. Counterexamples to the analogous conjectures for the spaces M g,n and M ct g,n have been constructed in [Petersen and Tommasi 2014;Petersen 2013]. The conjecture that Pixton's extension of the Faber-Zagier relations give rise to all relations in the tautological rings would imply that R • (M rt g,n ) fails to have Poincaré duality in general [Pixton 2013;Pandharipande, Pixton, and Zvonkine 2015;Janda 2013]; as would Yin's conjecture that all relations on the symmetric power C…”