  Let $(X, H)$ be a normal complex projective polarized variety and $\mathscr
E$ an $H$-semistable sheaf on $X$. We prove that the restriction $\mathscr
E\big|_C$ to a sufficiently positive general complete intersection curve $C
\subset X$ passing through a prescribed finite set of points $S \subset X$
remains semistable, provided that at each $p \in S$, the variety $X$ is smooth
and the factors of a Jordan-H\"older filtration of $\mathscr E$ are locally
free. As an application, we obtain a generalization of Miyaoka's generic
semipositivity theorem.
