Poincar\'e duality of wonderful compactifications and tautological rings

TL;DR

This paper establishes a deep connection between the Poincaré duality properties of tautological rings of certain moduli spaces of curves and their fibered powers, confirming a conjecture and extending to wonderful compactifications.

Contribution

It proves that Poincaré duality of tautological rings for $M_{g,n}^{rt}$ and $C_g^n$ are equivalent and provides a presentation of the tautological ring of $M_{g,n}^{rt}$ over that of $C_g^n$, confirming Tavakol's conjecture.

Findings

Poincaré duality holds for $M_{g,n}^{rt}$ iff it holds for $C_g^n$.

Presented the tautological ring of $M_{g,n}^{rt}$ as an algebra over that of $C_g^n$.

Results extend to the setting of wonderful compactifications.

Abstract

Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincar\'e duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.