On the structure of the kappa-ring

Abstract

We obtain lower bounds on the rank of the kappa ring of the Delign-Mumford compactification of the moduli space of curves in different degrees. For this purpose, we introduce a quotient of the kappa ring, the combinatorial kappa ring, and show that the rank of this latter ring in degree $d$ is bounded below by $|P(d,3g-2+n-d)|$ where $P(d,r)$ denotes the set of partitions of the positive integer $d$ into at most $r$ parts. In codimension 1 (i.e. $d=3g-4+n$) we show that the rank of the kappa ring is equal to $n-1$ for $g=1$, and is equal to $\lceil \frac{(n+1)(g+1)}{2}\rceil -1$ for $g>1$. Furthermore, in codimension $e=3g-3+n-d$, the rank of the kappa ring (as $g$ and $e$ remain fixed and $n$ grows large) is asymptotic to $\frac{\left(\genfrac{}{}{0ex}{}{n+e}{e}\right)\left(\genfrac{}{}{0ex}{}{g+e}{e}\right)}{(e+1)!}$.

In the revised version a number of theorems describing the structure of the kappa ring in codimension one are added to the original submission. Moreover, a theorem describing the asymptotic behaviour of the rank of the combinatorial kappa ring is added in the revision

References in corpus (1)

• The kappa ring of the moduli of curves of compact type: II