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**Extra info for A brill - noether theory for k-gonal nodal curves**

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Which we w r i t e as i cZ , to specify i t Call t h i s second s u b d i v i s i o n , ( i . e . as the f i r s t s u b d i v i s i o n of c~') C . Let ~ denote the s i m p l i c i a l r e g u l a r L L L,* neighborhood of the cone p o i n t in t h i s complex. Let ~ be the L,o r e g u l a r neighborhood of the barycenter b of the simplex o in ~L" We i d e n t i f y these spaces with t h e i r homeomorphic images in . Now we l e t ~ : L,J L L

1 2. Formal l i n k s and the PL Grassmannian We begin our discussion O f ~ n , k formal l i n k of dimension by i n t r o d u c i n g the notion of (n,k;j). If n and merely r e f e r to the dimension of the l i n k as Let U n+k R space be a /~n,k k are understood, we j. j + k - d i m e n s i o n a l subspace of the standard Euclidean S denotes the u n i t sphere in U (centered a t the U o r i g i n ) and D the u n i t d i s c . If zJ-Ic__, s is a t o p o l o g i c a l U U j-1 ( j - 1 ) - s p h e r e , an a d m i s s i b l e t r i a n g u l a t i o n i s a t r i a n g u l a t i o n of (as a c o m b i n a t o r i a l m a n i f o l d ) such t h a t : (a) r-simplex n+k dimensional subspace of R (b) n+k R , For e a c h If c(o) then (c) ~ is is o of containing the convex h u l l c(~) In p a r t i c u l a r , this o Definition.

Having d e f i n e d all each of CL. CL G But i n respects "o", for on v e r t i c e s so d o i n g , ~ CZLC L homeo "*" and n +k R "L " for of "L") C , we m e r e l y e x t e n d l i n e a r l y L we n o t e t h a t t h i s d e f i n i t i o n of G the i d e n t i f i c a t i o n s on was formed as ,k the u n i o n , mod i d e n t i f i c a t i o n s , of c e l l s c~ = IC I . 9 map G: /•n It is n+k + R ,k i m p o r t a n t to note t h a t G(~ ) C_V for all L. " n,k where TV denotes That i s , f i r s t define Yn,kJeL as (GIeL)*TV L, L the t a n g e n t PL n - b u n d l e o f the PL m a n i f o l d V .