Maximal degree subsheaves of torsion free sheaves on singular projective curves

Fix integers r, k, g with r > k > 0 and g greater than or equal to 2. Let X be an integral projective curve with g := p(a)(X) and E a rank r torsion free sheaf on X which is a flat limit of a family of locally free sheaves on X. Here we prove the existence of a rank k subsheaf A of E such that r(deg(A)) greater than or equal to k(deg(E)) - k(r - k)g. We show that for every g greater than or equal to 9 there is an integral projective curve X, X not Gorenstein, and a rank 2 torsion free sheaf E on X with no rank 1 subsheaf A with 2(deg(A)) greater than or equal to deg(E) - g. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.