Hilbert-kunz Multiplicity and Reduction Mod P

In this paper, we study the behaviour of Hilbert-Kunz multiplicities (abbreviated henceforth to HK multiplicities) of the reductions to positive characteristics of an irreducible projective curve in characteristic 0. For instance, consider the following question. Let f be a nonzero irreducible homogeneous element in the polynomial ring Z[X1, X2, . . . , Xr], and for any prime number p ∈ Z, let Rp = Z/pZ[X1, X2, . . . , Xr]/(f) (this is the homogeneous coordinate ring of a projective variety over Z/pZ)). Let HK(Rp) denote the Hilbert-Kunz multiplicity of Rp with respect to the graded maximal ideal. Then one can ask: does limp→∞HK(Rp) exist? This question was first encountered by the author in a survey article [C], Problem 4, section 5 (see also Remark 4.10 in [B1]). This seems a difficult question in general, as so far, there is no known general formula for HK multiplicity in terms of ‘better understood’ invariants. There does not seem to even be a heuristic argument as to why the limit should exist, in general, in arbitrary dimensions. However in the case of a projective curve (equivalently 2 dimensional standard graded ring) over an algebraically closed field of characteristic p > 0, one can express HK multiplicity in terms of (i) “standard” invariants of the curve which are constant in a flat family and (ii) normalized slopes of the quotients occuring in a strongly semistable Harder-Narasimhan filtration (HN filtration) of the associated vector bundle on the curve (see [B1] and [T1]). Hence, we may pose the question in the following more general setting. Given a projective curve X defined over a field k of char 0 with a vector bundle V on X of rank r, there exists a finitely generated Z-algebra A, a projective Ascheme XA such that XA ⊗Q(A) k = X, and coherent, locally free sheaves VA and E1A, . . . , ElA on XA such that, for all closed points s ∈ Spec A, if Vs = VA ⊗k(s), and Ei(s) = EiA ⊗ k(s), then 0 ⊂ E1(s) ⊂ · · · ⊂ El(s) ⊂ Vs is the HN filtration of Vs (we will give a detailed version of this in section 2). Choose st ≥ 0 such that