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Lemma 1.1. L e t Y b e a non-singular projectiv e v ariety w ith a v ery am ple inv ertible sheaf 6, (l ) an d let E , (o r, £ 2 ) be a stable (or, sem i-stable, resp.) sheaf Hence E is sem i-stable. N ote th a t if E is semi-stable and if E " is a coherent quotient sheaf of E with PE (m )=P,,(m ), then E" is torsion free. Then the proof of the converse is sim ilar to the above and easier, and hence we omit it.The following notion is originally due to C. S. Seshadri ([18] and [5]). Definition 1.5. S eim i-stable sheaves E ,, E 2 o n a non-singular projective variety are said to be S-equivalent if g r(E 1 ) is isomorphic to gr(E 2 ).Corollary 1.4.1 implies that every semi-stable sheaf is S-equivalent to one which is isomorphic to a direct sum of stable sheaves.Remark 1.6. 1) A stable sheaf E , is S-equivalent to E 2 if and only if E , is isomorphic to E2.') The center of G L(V) acts trivially on P(V, r, W ) . Thus PGL(V)acts on P(V, r, W ) . Though the o(1) may not be PGL(V)-linearized, 0(m) carries a PGL(V)-linearization for some positive integer m.

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