The connection between $R^{⁎}({\mathcal{C}}_g^n)$ and $R^{⁎}({\mathcal{M}}_{g,n}^{rt})$

Abstract

We study the connection between tautological classes on the moduli spaces ${\mathcal{C}}_g^n$ and ${\mathcal{M}}_{g,n}^{rt}$. A conjectural connection between tautological relations on ${\mathcal{C}}_g^n$ and ${\mathcal{M}}_{g,n}^{rt}$ is described. The analogue of this conjecture for the Gorenstein quotients of tautological rings is proved.

Introduction

Let ${\mathcal{M}}_{g,n}^{rt}$ be the moduli space of stable n-pointed curves of genus $g>1$ with rational tails. This space is a partial compactification of the space ${\mathcal{M}}_{g,n}$, classifying smooth curves of genus g together with n ordered distinct points. It sits inside the Deligne–Mumford compactification ${\bar{\mathcal{M}}}_{g,n}$ of ${\mathcal{M}}_{g,n}$, which classifies all stable n-pointed curves of genus g. We also consider the space ${\mathcal{C}}_g^n$ classifying smooth curves of genus g with not necessarily distinct n ordered points. There is a natural proper map from ${\mathcal{M}}_{g,n}^{rt}$ to ${\mathcal{C}}_g^n$ which contracts all rational components. Tautological classes on these spaces are natural algebraic cycles reflecting the geometry of curves. Their definition and analysis on ${\mathcal{M}}_g$ and ${\bar{\mathcal{M}}}_g$ was started by Mumford in the seminal paper [5]. These were later generalized to pointed spaces by Faber and Pandharipande [2]. In this short note we study the connection between tautological classes on ${\mathcal{C}}_g^n$ and ${\mathcal{M}}_{g,n}^{rt}$. We show that there is a natural filtration on the tautological ring of ${{\mathcal{M}}_{g,n}}^{rt}$ consisting of $g-2+n$ steps. A conjectural dictionary between tautological relations on ${\mathcal{C}}_g^n$ and ${{\mathcal{M}}_{g,n}}^{rt}$ is presented. We prove the analogue version of our conjecture for the Gorenstein quotients of tautological rings. It is well-known that the tautological group of ${\mathcal{M}}_{g,n}^{rt}$ is one dimensional in degree $g-2+n$ and it vanishes in higher degrees. In [3] the authors predicted that these should be deductible from the main theorem of Looijenga [4]. We show that both statements follow from the result of Looijenga on ${\mathcal{C}}_g^n$ and a result of Faber.

Conventions 0.1

We consider algebraic cycles modulo rational equivalence. All Chow groups are taken with $\mathbb{Q}$-coefficients.