J. London Math. Soc. (2) 74 (2006) 59–74 C2006 London Mathematical Society doi:10.1112/S0024610706022836 MAXIMAL SUBSHEAVES OF TORSION-FREE SHEAVES ON NODAL CURVES

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. 1