## “School of Mathematics”

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Paper   IPM / M / 16145
School of Mathematics
Title:   On the structure of the kappa-ring
Author(s):
 1 Eaman Eftekhary 2 Iman Setayesh
Status:   Published
Journal: Int. Math. Res. Not.
Vol.:  2017
Year:  2017
Pages:   3281-3321
Supported by:  IPM
Abstract:
We obtain lower bounds on the rank of the kappa ring \kring of the Delign-Mumford compactification of the moduli space of curves in different degrees. For this purpose, we introduce a quotient κc*(\Mgnbar) of \kring, and show that the rank of this latter ring in degree d is bounded below by |\PP(d,3g−2+nd)| where \PP(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 κc*(\Mgnbar) is equal to n−1 for g=1, and is equal to
 ⎡⎢ (n+1)(g+1) 2 ⎤⎥ −1
for g > 1. Furthermore, in codimension e=3g−3+nd, the rank of κc*(\Mgnbar) (as g and e remain fixed and n grows large) is asymptotic to
 ⎛⎝ n+e e ⎞⎠ ⎛⎝ g+e e ⎞⎠

(e+1)!
.