The rationality of the Hilbert-Kunz multiplicity in graded dimension two

We show that the Hilbert-Kunz multiplicity is a rational number for an R_±primary homogeneous ideal I=(f_1, …, f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f_1, . . ., f_n) on the projective curve Y = Proj R.