Hilbert–Kunz functions for irreducible plane curves

Let C be a characteristic p irreducible projective plane curve defined by a degree d form f, and n→en(f) be the Hilbert–Kunz function of f. en=μp2n−Rn with and Rn=O(pn). When C is smooth, Rn=O(1); Brenner has shown the Rn to be eventually periodic when one further assumes C defined over a finite field. We generalize these results, dropping smoothness. An additional term, (periodic) pn now appears in Rn, with the periodic function taking values in . We describe it using 1-dimensional Hilbert–Kunz theory in the local rings of the singular points of C, together with sheaf theory on C, and work explicit examples.