Hilbert - Kunz Multiplicity

1. An estimate for the HK multiplicity of a curve Let X be a nonsingular projective curve over an algebraically closed field k of characteristic p > 0. We fix the following notations for a vector bundle V on X . If 0 = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Ft ⊂ Ft+1 = V is the Harder-Narasimhan filtration (or HN filtration) then we denote μ(Fi) = μ(Fi/Fi−1) and μ(Fi) = μ(Fi+1/Fi). Now throughout the section we fix a vector bundle V of rank r with its HN filtration 0 = Ẽ0 ⊂ Ẽ1 ⊂ Ẽ2 ⊂ · · · ⊂ Ẽl ⊂ Ẽl+1 = V. Let μi = μ(Ẽi+1/Ẽi) and let μ = μ(V ), then the definition of HN filtration implies that μ1 > μ2 · · · > μl+1 and for some 1 ≤ i ≤ l we have μi ≥ μ ≥ μi+1. Lemma 1.1. Suppose the characteristic p satisfies p > 4(g − 1)r!. Then F Ẽ1 ⊂ F Ẽ2 ⊂ · · · ⊂ F Ẽl ⊂ F V is a subfiltration of the HN filtration of F V . Proof. For each 0 ≤ i ≤ l + 1, let F Ẽi ⊂ Ei1 ⊂ · · · ⊂ Eiti ⊂ F Ẽi+1 be a filtration of vector bundles on X such that 0 ⊂ Ei1/F Ẽi ⊂ Ei2/F Ẽi ⊂ · · · ⊂ F Ẽi+1/F Ẽi is the HN filtration of F (Ẽi+1/Ẽi). Then by [SB], we have (1.1) 0 ≤ μmaxF (Ẽi+1/Ẽi)− μminF (Ẽi+1/Ẽi) ≤ (2g − 2)(r − 1). Since μmaxF (Ẽi+1/Ẽi) ≥ μ(F (Ẽi+1/Ẽi)) ≥ μminF (Ẽi+1/Ẽi), we have (1.2) 0 ≤ μ(Ei1/F Ẽi)− μ(F (Ẽi+1/Ẽi)) ≤ (2g − 2)(r − 1)