Maximal degree subsheaves of torsion free sheaves on singular projective curves

Fix integers $r,k,g$ with $r>k>0$ and $g\ge 2$. 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 $r(\deg(A))\ge k(\deg (E))-k(r-k)g$. We show that for every $g\ge 9$ 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 $2(\deg (A))\ge \deg(E)-g$. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.