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**Read Online or Download Algebraic Geometry Sundance 1986 PDF**

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**Additional info for Algebraic Geometry Sundance 1986**

**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.