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H°(S,@s(D-Fi)) -~ H°(S,E(-Fi)) --~ H°(S,@s(A-D-Fi) --~ H~(S,(Ss(D-Fi)). A < 0, the first space is zero. Since (D-Fi) 2 = - 4 , by R i e m a n n - R o c h , the last space is o n e dimensional. This implies that H°(S,E(-Fi)) :# O, hence E contains (gs(Fi) as a subsheaf, and therefore is represented as an extension 0--* (Ss(Fi) --o E --~ (gS(A-Fi) --o 0. A = 7 > 5. This contradicts the s e m i stability of E. By Theorem 2, E must be the Reye bundle.

4. Stability. Recall that a vector bundle E i s H - s t a b l e (resp. H-semi-stable), where H is a divisor, if for every line subbtmdle L in E L-H < ½ q ( E ) - H (resp. L,H _<½cl(E)-H). Theorem 3. Let E be a rank 2 vector bundle on an Enriques surface S with q ( E ) = A and ~ ( E ) = 3. The following assertions are equivalent: (i) E is A-semi-stable; (ii) E is isomorphic to the Reye bundle. (iii) E is A-stable. PROOF. (i) ~ (ii) By Proposition 1, h°(E) ;~ 0. We may assume that the zero set of any non-zero section 46 has a l - d i m e n s i o n a l part D.

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