On the Harder-Narasimhan Filtration for Coherent Sheaves on P: I

Let E be a torsion-free sheaf on P. We give an effective method which uses the Hilbert function of E to construct a weak version of the Harder-Narasimhan filtration of a torsionfree sheaf on P subject only to the condition that E be sufficiently general among sheaves with that Hilbert function. This algorithm uses on a generalization of Davis’ decomposition lemma to higher rank. Consider the following problem. Let E be an explicit torsion-free sheaf on P2 given by a presentation 0 → ⊕ n∈Z OP2(−n) b(n) φ → ⊕ n∈Z OP2(−n) a(n) → E → 0. (0.1.1) How does one go about effectively computing the Harder-Narasimhan filtration of E , i.e. the unique filtration 0 = F0(E) ⊂ F1(E) ⊂ · · · ⊂ Fs(E) = E such that the graded pieces gri(E) := Fi(E)/Fi−1(E) are semistable in the sense of GiesekerMaruyama and their reduced Hilbert polynomials Pi(n) = χ(gri(E)(n))/rk(gri(E)) satisfy P1(n) > P2(n) > · · · > Ps(n) for n ≫ 0? In this paper and its planned sequel we consider the problem under the simplifying assumption that the matrix φ of homogeneous polynomials is general, i.e. that E is general among torsion-free sheaves with the same Hilbert function as E . Our solution to the problem then divides into two parts. In this first part we construct a filtration of E of the type 0 ⊂ E≤τ1 ⊂ · · · ⊂ E≤τs ⊂ E (0.1.2) where E≤n denotes the subsheaf of E which is the image of the natural evaluation map H0(E(n)) ⊗ OP2(−n) → E . We give an algorithm for picking the τi so that the filtration approximates the true Harder-Narasimhan filtration but groups together all pieces of the Harder-Narasimhan filtration with slopes between two consecutive integers. The associated graded sheaves gri(E) := E≤τi/E≤τi−1 are not always semistable, but they do share a number ∗Supported in part by NSA research grant MDA904-92-H-3009.