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

Abstract

Let ${\kappa }_e({\overline{M}}_{g,n})$ denote the kappa ring of ${\overline{M}}_{g,n}$ in codimension $e$. For $g,e\ge 0$ 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 $$\frac{\left(\genfrac{}{}{0ex}{}{n+e}{e}\right)\left(\genfrac{}{}{0ex}{}{g+e}{e}\right)}{(e+1)!}\simeq \frac{\left(\genfrac{}{}{0ex}{}{g+e}{e}\right)n^e}{e!(e+1)!}.$$ When $g\le 2$ 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 $i$.

References in corpus (2)

• The structure of the tautological ring in genus one
• On the structure of the kappa-ring