The tautological ring of the space of pointed genus two curves of compact type

Abstract

We prove that the tautological ring of $M_{2,n}^{ct}$, the moduli space of n-pointed genus two curves of compact type, does not have Poincaré duality for any $n\ge 8$. This result is obtained via a more general study of the cohomology groups of $M_{2,n}^{ct}$. 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 $H^k(M_{2,n}^{ct})$ for any $k$ and $n$ considered both as $S_n$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of ${\overline{M}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of ${\overline{M}}_{2,n}$ fails to have Poincaré duality for all $n\ge 20$. This improves and simplifies results of the author and Orsola Tommasi.

23 pages. Major revision, includes new results on moduli of stable genus two curves and a proof of a conjecture left open in the author's paper with Orsola Tommasi. Paper is reorganized, some proofs simplified

References in corpus (1)

• Cohomology of local systems on the moduli of principally polarized abelian surfaces

Cited by in corpus (6)

• Nontautological Bielliptic Cycles
• Extremality of loci of hyperelliptic curves with marked Weierstrass points
• Extensions of tautological rings and motivic structures in the cohomology of ${\overline{\mathcal{M}}}_{g,n}$
• The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$
• The tautological ring of ${\overline{\mathcal{M}}}_{g,n}$ is rarely Gorenstein
• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields