SPECIAL DIVISOR S O N ALGEBRAI C CURVE S 1 3 symmetric produc t C d is a complex manifol d o f dimension d, an d in Cd we defin e th e sub- varieties Crd={DGCd.r(D)r}. Since, according to the geometri c Riemann-Roc h theore m (1.33 ) (1.34) r(D) r if, and onl y if dim D d - 1 - r, it is clear tha t C d is in fac t a closed su b variety. Recallin g ou r basi s col, ..., OJ for the space of holomorphic differentials , w e may expres s the conditio n in (1.34) algebraically b y introducing th e Brill-Noether matrix Ui(Pd)'"°g(Pd)/ The rth row of K(D) i s the homogeneous coordinat e vecto r of PK(pi) G P^_ 1 . (I f some of the pt coincide-e.g., if px = p2-then w e take th e secon d ro w to be any poin t othe r tha n PK(Pi) o n the tangen t lin e to PK(C) at pt. Thi s is consistent wit h th e comple x manifol d structure o n the symmetri c produc t C d around th e variou s diagonals. ) Now, it is clear o n the on e hand tha t dim D = rank n(D) - 1, and consequentl y (1.34 ) ma y be rephrased as (1.36) r(D) r if, and onl y if rank n(D)d-r. On the othe r hand , it is wel l known tha t i n the spac e of all g x d matrices K the determinan - tal variet y rank K d - r has pure codimensio n r(g - d + r). Consequentl y w e obtain : (1.37) di m Crdd-r(g-d + r). More precisely, for any irreducible componen t o f Cd th e inequalit y (1.37 ) is valid. (I f in ad- dition, a suitable transversalit y conditio n is satisfied the n w e will have equality in (1.37).) Finally, since for any line bundle L with h°(L) = r -f 1 there will be a Pr of divisors/) G |Z,|, if w e introduce th e varietie s (th e variet y structur e wil l be give n in the nex t section ) H/J={La lin e bundle of degree d with h°(L)r+ 1} , then (1.37 ) implies tha t dim W d d - r(g - d + r) - r = g - (r + \)(g - d + r ). (We are omittin g certai n technica l point s tha t ar e explained in detail in [1].)
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