Generalized Hilbert-Kunz function in graded dimension 2

We prove that the generalized Hilbert-Kunz function of a graded module over a two-dimensional standard graded normal -domain over an algebraically closed field of prime characteristic has the form , with rational generalized Hilbert-Kunz multiplicity and a bounded function. Moreover, we prove that if is a -algebra, the limit for of the generalized Hilbert-Kunz multiplicity over the fibers exists, and it is a rational number.