By (2.2) and (2.4), imrf has dimension (L2 Cs__x)-At + 1 g[ and codimen-
sion Lx - Ar So im rf is precisely the subspace of
T(An 0(L, + L2 - C
_,)/0(L, + L2 - C,))
of elements which vanish on At to the order given by (/) , the divisor of/. There
are thus Lx -At vanishing conditions on imr^ as a subspace of
T(A„ 0(L, + L2 - C
_,)/0(^i + L2 - C,)).
By Proposition 2.1, im T D im Ty.
Now consider the first Ct such that At C supp(C/), i.e. v4, (£supp(C,_}) and
Ct Ct_x -\r Ar Let L be a line bundle on M. Then, on An sections of L Ci_l
may be identified with those sections of L on At which vanish at supp(Q_!) D At
in prescribed ways. By construction, supp(C/_1) D At n supp(/) = 0 .
T(A„ 0(L, - C,_,)/0(L, - C,)) ®
r ( ^ 0(L
- C,_,)/0(L
- C,))
- r ( , 4 0(L, + L2 - C,_, - C
_,)/0(L, +L2- C,_, - C,_, - A,))
^T(A„ 0(L, + L2 - C,_,)/e(L, + L2 - C,)),
whjere a multiplies sections and t is the inclusion map described just above. To
prove (3.1), it suffices that im(t o «) contain elements with any given prescribed
zeros strictly less than (/) . All of the prescribed zeros are in supp(/). The total
number of prescribed zeros, counting multiplicities, is at most Lx - At 1. This is
essentially a vanishing condition on im a.
By (2.2), in order for a section to have prescribed zeros of total order r in a line
bundle L, it suffices that c(L) 2g' + r. The condition c(L) 2gf + r also
allows the section to be nonzero at any finite number of prescribed points. So
multiplying prescribed sections will add prescribed zeros. In a of (3.3), we may
allocate, as follows, the L, -At 1 vanishing conditions between
T(A„e(L]-Cl_l)/e(L{-Cl)) and T{A„Q{L2-CS_,)/Q{L2-Cs)).
In case LrAt0 and L2A, 0, by (2.4), (L, - C^^-A, s* 2g', and
(L2 Cs_x)-A, 2g'r So it suffices for the allocation that
(3.4) (L, - C,_x)-A, + (L2 - Cs_x)-At 4g', + LrA,-l.
Since L2At 3KAt with strict inequality in the case KAt = 0, L2At^
2KAt + 1. Then (3.4) follows from (2.4).
The remaining case is that Lx At 0 or L2At 0. For Lx -At 0, / is
nonzero on At and ( / ) gives no vanishing conditions. The map rf of (3.2) is
already onto. For L2 - At 0, reverse the roles of Lx and L2. Again ( / ) gives no
vanishing conditions and Tf of the new (3.2) is onto. So (3.1) is true for all Cs and
so T is surjective.
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