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Download Algebraic Geometry Sundance 1986 by Holme R. Speiser (Eds.) PDF

By Holme R. Speiser (Eds.)

This quantity provides chosen papers because of the assembly at Sundance on enumerative algebraic geometry. The papers are unique learn articles and focus on the underlying geometry of the topic.

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Example text

Let B i be the base of the etale v e r s a t d e f o r m a t i o n space for t h e s i n g u l a r i t y of C a t Pi. F r o m the d e f o r m a t i o n t h e o r y of [D-H2] w e see t h a t (after etale base change) a neighborhood of q in pN m a p s to t h e product of the spaces 1:5i and n e a r the origin (0 . . . 0) the m a p is s u r j e c t i v e w i t h s m o o t h fibers. 4), finishes t h e proof of t h e f o r m u l a s for r ( C U ) , r ( T N ) a n d r(TR). The c o m p u t a t i o n of r (NL) likewise reduces to an e x a m i n a t i o n of local d e f o r m a t i o n t h e o r y , in this case t h e condition for a first order d e f o r m a t i o n of a c u r v e C h a v i n g a node a t a point p on a line L to p r e s e r v e the node a n d keep it on L.

For example, we once m o r e h a v e a universal flat f a m i l y n : ~ - + W of c u r v e s of a r i t h m e t i c genus g over W, whose fiber over C E W is the n o r m a l i z a t i o n of t h e corresponding plane c u r v e at its assigned nodes (an "assigned" node m a y be defined to be a limit of nodes of c u r v e s Cx £ W lying over V c V and tending to C; t h u s for C ¢ A the fiber of ~ will be the n o r m a l i z a t i o n of C, while for C ¢ A it will be the n o r m a l i z a t i o n of C at all the 26 nodes except for the one corresponding to the sheet of W containing C).

T 1 = s, t 2 = r, t~ = r, t4 = r s or in C a r t e s i a n f o r m b y e q u a t i o n s : 1. t 3 = 0, t4 = 0 2. t 2 = 0, t4 = 0 3. t 2 = t3, t I t 2 = t 4. o b s e r v e t h a t w h e n w e pull t h e s e loci b a c k to t h e ( r , s ) - p l a n e , t h e l o c u s of c u r v e s w i t h t h r e e n o d e s is g i v e n in b r a n c h 1) b y r = 0, t h e locus of c u r v e s w i t h a t a c n o d e b y s 2 = 4r; and that these have intersection multiplicity b r a n c h 2) t h e s e t w o loci a r e g i v e n b y t h e e q u a t i o n s respectively, and have intersection number s2 = - 4 r , s2 = 4r 2; a n d in b r a n c h 3) b y r -- 0 a n d again having intersection multiplicity m u l t i p l i c i t y of t h e s e t w o loci is t h u s r = 0 and 2; s i m i l a r l y in 2.

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