Bounds on the Castelnuovo-Mumford Regularity in dimension two

Consider a Cohen-Macaulay local ring (R, m) with dimension d 2, and let I R be an m-primary ideal. Denote r₉ (I) as the reduction number of I with respect to a minimal reduction J of I, and (I) as the stability index of the Ratliff-Rush filtration with respect to I. In this paper, we derive a bound on (I) in terms of the Hilbert coefficients and r₉ (I). In the case of two-dimensional Cohen-Macaulay local rings, the established bound on (I) consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of I.