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Paper   IPM / M / 16146
School of Mathematics
Title:   On the kappa ring of Mg,n
Author(s):
 1 Eaman Eftekhary 2 Iman Setayesh
Status:   Published
Vol.:  298
Year:  2016
Pages:   89-121
Supported by:  IPM
Abstract:
Let κe(\Mgnbar) denote the kappa ring of \Mgnbar in codimension e (equivalently, in degree d=3g−3+ne). For g,e ≥ 0 fixed, as the number n of the markings grows large we show that the rank of κe(\Mgnbar) is asymptotic to
 ⎛⎝ n+e e ⎞⎠ ⎛⎝ g+e e ⎞⎠

(e+1)!
 ⎛⎝ g+e e ⎞⎠ ne

e!(e+1)!
.
When g ≤ 2 we show that a kappa class κ ∈ \kring 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 κnd(\Mbar1,n) is equal to |\PP1(d,nd)|, where \PPi(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.