On the kappa ring of $\overline{M}_{g,n}$

TL;DR

This paper investigates the structure and rank of the kappa ring of the moduli space of stable curves, providing asymptotic formulas and characterizations for low genus cases.

Contribution

It derives asymptotic formulas for the rank of the kappa ring as the number of markings grows and characterizes trivial kappa classes for low genus cases.

Findings

Rank of kappa ring asymptotically proportional to n^e for large n.

A kappa class is trivial if and only if its integrals against all boundary strata vanish for g ≤ 2.

Exact rank formula for g=1 involving partition sets P_1(d,n-d).

Abstract

Let $\kappa_e(\overline{M}_{g,n})$ denote the kappa ring of $\overline{M}_{g,n}$ in codimension $e$. For $g,e\geq 0$ fixed, as the number $n$ of the markings grows large we show that the rank of $\kappa_e(\overline{M}_{g,n})$ is asymptotic to $$\frac{{n+e\choose e}{g+e\choose e}}{(e+1)!}\simeq \frac{{g+e\choose e}n^e}{e!(e+1)!}.$$ When $g\leq 2$ we show that a kappa class $\kappa$ is trivial if and only if the integral of $\kappa$ against all boundary strata is trivial. For $g=1$ we further show that the rank of $\kappa_{n-d}(\overline{M}_{1,n})$ is equal to $|P_1(d,n-d)|$, where $P_i(d,k)$ denotes the set of partitions $p=(p_1,...,p_\ell)$ of $d$ such that at most $k$ of the numbers $p_1,...,p_\ell$ are greater than $i$.