By Audun Holme, Robert Speiser

ISBN-10: 3540192360

ISBN-13: 9783540192367

This quantity provides chosen papers due to the assembly at Sundance on enumerative algebraic geometry. The papers are unique study articles and focus on the underlying geometry of the subject.

**Read Online or Download Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986 PDF**

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**Additional resources for Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986**

**Example text**

To prove the second statement, fix 5 as above. It suffices to show that a general element of V induces a section of the line bundle JIC/S ~ having 60 only simple zeros. But b y o u r hypothesis V g e n e r a t e s JlC/S ~, so again b y a version of Bertini's T h e o r e m a general section v a n i s h e s on a reduced set of points, as desired. Finally, t h e f o r m u l a s for t h e n u m e r i c a l c h a r a c t e r s of C' a n d COC' follow a t caace if w e p u t t o g e t h e r t h e facts t h a t , b e c a u s e D is a c o m p l e t e intersection of r - 1 quadrics, its canonical bundle is t h e restriction of {3pr(r-3), a n d t h e r e s t r i c t i o n of its canonical bundle to C or C' is t h e canonical bundle of t h e s m a l l e r c u r v e twisted b y t h e divisor on t h a t c u r v e w h i c h is t h e s u m of t h e points of CAC'.

On the other hand, the principal fails already for some non projectively normal curves in p5. The example of smallest degree is the general elliptic octic in pS, which is, as w e show, cut out scheme theoretically by 5 quadric~, though its homogeneous ideal requires two additional cubic generators (section 5). The example is constructed, following the attack of Mori [1984], by exploiting the surjectivity of the period m a p for K3 surfaces to first construct the K3 surface in p5 which will be the intersection of 3 general quadrics containing C.

This will give a f a m i l y of plane c u r v e s of degree 3 m - Ill and genus 1, with divisor D given as above. To c o m p u t e the class co note t h a t the canonical bundle of ~2 blown up at 9 points is - 3 H + E 1 . . + E 9 . We m u s t s u b t r a c t f r o m this the pullback of the canonical bundle of t h e base of the family, which is m i n u s two fibers; we h a v e co = - 3 H + E i + ... E 9 - ( - 2 ( 3 H - EX - ... -Eg. Thus on the base of the f a m i l y we h a v e deg(A) : m 2-1II deg(B) = 3 m - I I I deg(C) = O.