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By Grauert G. (ed.)

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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. PROOF. It is clear that any nonsingular curve CE[A[ is of genus 6. It is easy to see that C is not hyperelliptic (see [CD1]). Assume C is trigonal. Then its canonical image lies on a scroll, hence C has infinitely many (ool) trisecants.

The vector bundle E constructed in the proof of the previouis theorem is called the Reye bundle. Obviously, q(E) = A is the Reye polarization, and c2(E) = 3. Since E is the restriction of the universal quotient bundle of G(2,4) to S, its isomorphism class is independent of the choice of F i. P r o p o s i t i o n I. Let E be a rank 2 vector bundle on S with c 1(E) = ~ and c2(E) = 3. Then h°(E) >-4. PROOF. By Riemann-Roch: hO(E)+h°(E*(K)) = 4+h~(E). If h°(E*(K)) = 0, the assertion is obvious. Assume h°(E*(K)) ~ 0.

C is a line. C = 1 and (A-Fi-C) 2 = 6. By R i e m a n n - R o c h , dim IA-Fi-C] >_3 which is absurd. To show that IA-Fit has no isolated base points, it is e n o u g h to verity that for every nef divisor F with F 2 = 0 one has (A-Fi)°F 2 2. ([CDll, Thin. 1). By R i e m a r m - R o c h , A - F i - F j i s effective if i * j . Thus (A-Fi)°F = (A-Fi-Fj)°F+F j °F _>FfF. If F°Fj > 1 for some j¢ i we are done. F = 9+F°F i, and (A-Fi)°F = 3 - 3F°Fi . F i = 3, AoF = 4. (F+Fi) 2 -(A-(F+Fi))-) = 6 0 - 4 9 > 0.

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Complex Analysis and Algebraic Geometry by Grauert G. (ed.)


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