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In this paper we show that the sets of $F$-jumping coefficients of idealsform discrete sets in certain graded $F$-finite rings. We do so by giving acriterion based on linear bounds for the growth of the Castelnuovo-Mumfordregularity of certain ideals. We further show that these linear bounds existsfor one-dimensional rings and for ideals of (most) two-dimensional domains. Weconclude by applying our technique to prove that all sets of $F$-jumpingcoefficients of all ideals in the determinantal ring given as the quotient by$2\times 2$ minors in a $2\times 3$ matrix of indeterminates form discretesets.

Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings