Algebraic Geometry: Proceedings of the U. S.-U. S. S. R. - download pdf or read online

By Spencer J. Bloch, I. Dolgachev, W. Fulton

ISBN-10: 0387544569

ISBN-13: 9780387544564

This quantity comprises the court cases of a joint USA-USSR symposium on algebraic geometry, held in Chicago, united states, in June-July 1989.

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Extra info for Algebraic Geometry: Proceedings of the U. S.-U. S. S. R. Symposium Held in Chicago, June 20-July 14, 1989

Sample text

Then S can he embedded by IAI into IP 5. A is said to be a Reye polarization if the image of S ties on a quadric. A is a said to be a Cavley polarization it" A+KS isa Reye polarization. e. contains smooth rational curves (nodalcurves). Note that every Enriques surface lying on a quadric is isomorphic to a Reye congruence o f lines inlP -~(char(k) ~ 2). The Cayley polarization maps S onto a surface in IP 5 isomorphic to the surface of reducible quadrics in a 5-dimensional linear system of quadrics in IP 5.

Ii) ~ (iii). This follows from the proof of T h e o r e m 1 (take a destabilizing s u b b u n d l e L a n d argue as in the proof o f this theorem). (iii) ~ (ii) Obvious. Examples. Here we give examples o f n o n - r e g u l a r rank 2 vector bundles on S with ct(E) = A and ca(E) = 3. The first example is a decomposable bundle E =Gs(D)~)(gS(A-D), where D = FI+F2+F 3. Then D 2= 6, D-(A-D) = 3, (A-D) 2 = - 2 , A-(A-D) = 1. If rA-Dt = O, Le. S does not contain lines (with respect to A), then 47 H°(E) = H°((gs(D)), hence alt sections o f E vanish on a curve from IDI.

This contradicts the s e m i stability of E. By Theorem 2, E must be the Reye bundle. C a s e Z D 2 = O. Then D-A = 3, hence D = F i for some i. Also deg(Z')' = O, and we get the exact sequence as above. Applying Theorem 2, we obtain that E is the Reye bundle. (ii) ~ (iii). This follows from the proof of T h e o r e m 1 (take a destabilizing s u b b u n d l e L a n d argue as in the proof o f this theorem). (iii) ~ (ii) Obvious. Examples. Here we give examples o f n o n - r e g u l a r rank 2 vector bundles on S with ct(E) = A and ca(E) = 3.

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Algebraic Geometry: Proceedings of the U. S.-U. S. S. R. Symposium Held in Chicago, June 20-July 14, 1989 by Spencer J. Bloch, I. Dolgachev, W. Fulton


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