On the number of components of the moduli schemes of stable torsion-free sheaves on integral curves

Here we give an upper bound for the number of irreducible components of the moduli scheme of stable rank $r$ torsion-free sheaves of fixed degree on the integral curve $X$. This bound depends only on $r$, $\mathrm {Sing}(X), p_a(X)$ and the corresponding number for the rank 1 case.