Algebraic Geometry Sundance 1986: Proceedings of a by Audun Holme, Robert Speiser PDF

By Audun Holme, Robert Speiser

ISBN-10: 3540192360

ISBN-13: 9783540192367

ISBN-10: 3540391576

ISBN-13: 9783540391579

This quantity provides chosen papers caused by 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 resources for Algebraic Geometry Sundance 1986: Proceedings of a Conference held at Sundance, Utah, August 12–19, 1986

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Q 2 - i ~ I E i to p r o j e c t i v e space u s i n g t h e l i n e a r s y s t e m IDI, t h e n t a k e a g e n e r i c projection of the image to p 2 This will give a f a m i l y of plane c u r v e s of degree 2a+Sb-lll and genus 2. q 2 - deE(A) = 2 a b - t l I deE(B) = 4b- 2 a - III ~ E. i I= 1 40 deg(C) = 28 deg(A) = 20. The proof of independence n o w follows i m m e d i a t e l y f r o m the existence of these families. Specifically, to show a) a b o v e use families i and 2; for b) use families 1, 2 and 3; for c) use families 1, 2, 4 and 5 a n d for d) use families i a n d 4.

El2 the o t h e r exceptional divisors; v i e w X as a f a m i l y of c u r v e s of genus 2 o v e r ~pl. F r o m the dimension calculations of [D-HI] we also see t h a t t h e general fiber of this f a m i l y is nonsingular a n d t h e singular fibers e a c h h a v e only one simple node as a singularity. Let I be a subset of {I, 2 . . . 12} containing a t m o s t 3 elements; again, denote b y III t h e n u m b e r of e l e m e n t s in I. 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.

4) below) one m a y see t h a t W' (d, 8) and W (d, 8) are both smooth. This allows us to identify Pic(W(d, 6)) and Pic(W'(d, 8)); call this identification j,. From standard intersection theory (See Fulton, [F]) we get a 30 homomorphism g~n 2 j . • Pic(W(d, 8)) ~ Pic(W(d, 8 + i)). 2) Definitipn: r = g, o n 2 oi x o j . To c o m p u t e t h e h o m o m o r p h i s m r explicitly we need a description of the local s t r u c t u r e of W'(d, 8) n e a r A'(d, 8) . By t h e d e f o r m a t i o n t h e o r y of [D-H2] we see t h a t we m a y obtain this local i n f o r m a t i o n b y looking in the d e f o r m a t i o n spaces of a p p r o p r i a t e singularities.

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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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