By Holme R. Speiser (Eds.)

This quantity provides chosen papers caused by 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**

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.

5) now allows us to c o m p u t e r(A) in two ways. They both come out to be equal to (8 + I ) A . 5). 6) Theorem: Let S(d, 8) c Pic(W(d, 8)) ® Q be the subspace spanned b y A, 2). Then the dimension of S(d, B,C, and A; a s s u m e t h a t 0<_ 8_< ~ ( d - l ) ( d 8) as a vector space over • is: i). dimS(d, O) = i ii). dimS(d, i) = 2 iii). dimS(d, 2) = 3 iv). dimS(d, 8) = 4 for 3 <_ 8 <_ ~(d-1)(d-2) - 2, v). e. g = 1), vi). e. g = 0). Proof: W(d, 0) is IPN with a set of codimension 2 r e m o v e d so clearly d i m S ( d , 0) = i.

T I = s, t 2 = r, t 5 = 0, t4 = 0 2. t 1 = s, t 2 = 0, t 3 = r, t4 = 0 as 32 3. 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.