Algebraic Geometry: Proceedings of the US-USSR Symposium by Spencer Bloch, Igor V. Dolgachev, William Fulton

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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.

Assume that A is a Cayley polarization. Then the union of trisecants of S is isomorphic tothe quartic hypersurface of singular quadrics in the 5-dimensional linear system of quadrics parametrized by [hi*. Corollary 3. Let S b e an Enriques surface o f degree 10 in IPS and C be its smooth hyperplane section. I f A = C3S(1) is not Reye, then C is a non-trigonal curve o f genus 6. I f A is Reye, then C is a trigonal curve o f genus 6 i f and only i f the hyperplane is tangent to the quadIic containing S.

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