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 s r ′ ( E ) = min s(E,F) , where the minimum is taken over all subsheaves of E of rank r ′. For a fixed r ′, s r ′ defines a stratification of the moduli space U(r,d) of stable torsion‐free sheaves of rank r , degree d by locally closed subsets U r ′, 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.