Hilbert–Kunz theory for nodal cubics, via sheaves

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n→en be the Hilbert–Kunz function of B with respect to J.Let q=pn. When J=(x,y,z), it is known that where if , and is 1 otherwise. We generalize this, showing that en=μq2+αq−R where R only depends on . We describe α and R in terms of classification data for a vector bundle on C.