By Holme R. Speiser (Eds.)
This quantity offers chosen papers caused by the assembly at Sundance on enumerative algebraic geometry. The papers are unique examine articles and focus on the underlying geometry of the topic.
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Additional resources for Algebraic Geometry Sundance 1986
Pro. 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.
Thus this n u m b e r m u s t be twice t h e degree of C~. The f o r m u l a s of Proposition 2 2 a n d simple a r i t h m e t i c n o w yield t h e de~sir~l r ~ u l t . Proof of Pro]x~ition 2 6 : First, let D : COC' be t h e c o m p l e t e intersection of quadrics. W r i t e ~C for t h e ideal sheaf of C in p r a n d s i m i l a r l y for C' a n d D. Since t h e canonical bundle on D is given b y OaD:OD(r-3), we h a v e b y t h e t h e o r y of liaison t h a t ~C/~D : (~D:~C')/~D : Hom((~C, , OEO = Hom(0c', ~D)(3-r) = ~C~3-r).
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.