Maximal subsheaves of torsion-free sheaves on nodal curves

Abstract

Let Y be a reduced irreducible projective curve of arithmetic genus g ≥ 2 with at most ordinary nodes as singularities. For a subsheaf F of rank r′, degree d′ of a torsion-free sheaf E of rank r, degree d, let s(E,F) = r′d-rd′. Define sr′(E) = min s(E,F), where the minimum is taken over all subsheaves of E of rank r′. For a fixed r′, sr′ defines a stratification of the moduli space U(r,d) of stable torsion-free sheaves of rank r, degree d by locally closed subsets Ur′,s. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank r′ for s ≤ r′(r-r′)(g − 1) (respectively s < r′(r-r′)(g − 1)). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve Y is nonspecial.