Download e-book for iPad: Algebraic Geometry Sundance 1986: Proceedings of a by Audun Holme, Robert Speiser

By Audun Holme, Robert Speiser

ISBN-10: 3540192360

ISBN-13: 9783540192367

ISBN-10: 3540391576

ISBN-13: 9783540391579

This quantity offers chosen papers as a result of the assembly at Sundance on enumerative algebraic geometry. The papers are unique study articles and focus on the underlying geometry of the subject.

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

Example text

H - ~E 1 i~l 39 w h e r e m_> 1 and I = ~ if m = 1; as in the previous case m a p X to projective space using the linear s y s t e m IDI and take a generic projection to p2. This gives a f a m i l y of plane c u r v e s of degree 4 m - III and genus 2. In a m a n n e r similar to Family Three we find that: ¢O = 5 H - 5E 0 : E l - . , . - E12 deE(A) = m 2 - III deE(B) = Sin- 111 deE(C) = 46 and deE(A) = 20. Family,Five. Consider a pencil of c u r v e s of t y p e (3,2) on ~ l x p1. One m a y easily check (by writing down equations) t h a t if t h e pencil is as general as possible the general e l e m e n t of this pencil will be nonsingular and the singular elements will each h a v e only one simple node as a singularity.

Finally for 8 z 3 a m a t r i x t h a t r e p r e s e n t s r: S(d, 8) - , S(d, 8+1) is o 8-1 0 6 8-2 0 0 w h i c h h a s r a n k 4. We conclude t h a t i f d i m S ( d , 8') = 4 for all 3_< 8'_< 8. d i m S ( d , $ ) = 4 for s o m e 8_> 3 t h e n P u t t i n g all we h a v e said t h u s f a r together one concludes t h a t to p r o v e the t h e o r e m it is sufficient to v e r i f y that: a). d i m S ( d , ~ ( d - 1 ) ( d - 2))_> 2 for d > 2 b). d i m S ( d , i ( d - 1 ) ( d - 2)) z 3 for d> 3 36 c).

Before we s t a r t our analysis, observe t h a t we h a v e the potential for a notational n i g h t m a r e here, with t h r e e s e p a r a t e varieties isomorphic to IP2 appearing in this picture: we h a v e of course the plane in w h l c h the c u r v e s of the net live, w h i c h we will denote b y p2; we h a v e the plane ,D p a r a m e t r i z i n g the c u r v e s in the net; a n d we h a v e the dual plane ~ = ~Dv w h i c h a p p e a r s n a t u r a l l y as the t a r g e t space of the m a p %0: p 2 _ , given b y the n e t JD.

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Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986 by Audun Holme, Robert Speiser


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