On the kappa ring of M‾g,n

Let κe(M‾g,n) denote the kappa ring of M‾g,n in dimension e (equivalently, in degree d=3g−3+n−e). For g,e≥0 fixed, as the number n of the markings grows large we show that the rank of κe(M‾g,n) is asymptotic to(n+ee)(g+ee)(e+1)!≃(g+ee)nee!(e+1)!. When g≤2 we show that an element κ∈κ⁎(M‾g,n) is trivial if and only if the integral of κ against all boundary strata is trivial. For g=1 we further show that the rank of κn−d(M‾1,n) is equal to |P1(d,n−d)|, where Pi(d,k) denotes the set of partitions p=(p1,…,pℓ) of d such that at most k of the numbers p1,…,pℓ are greater than i.