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Let    (  X  ,  H  )    be a normal complex projective polarized variety and   E   an  H ‐semistable sheaf on  X . We prove that the restrictionE  |   C    to a sufficiently positive general complete intersection curve    C  ⊂  X    passing through a prescribed finite set of points    S  ⊂  X    remains semistable, provided that at each    p  ∈  S   , the variety  X  is smooth and the factors of a Jordan–Hölder filtration of   E   are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.

A Mehta–Ramanathan theorem for linear systems with basepoints