SPECIAL DIVISOR S O N ALGEBRAI C CURVE S 13

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].)