symmetric produc t C
is a complex manifol d o f dimension d, an d in Cd we defin e th e sub-
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
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
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
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 - r(g - d + r) - r = g - (r + \)(g - d +
(We are omittin g certai n technica l point s tha t ar e explained in detail in [1].)
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