and effective divisors Fm on °V^, p supp Fm9 such that, letting F* denote the total
transform of Fm under II
(a) The T/ generate T(M, Q{mK - D)) over T(V9 0
(b) The T* have no common zeros.
(c) The map Of: : T*): M - P
/ _ 1
is (somewhere) nondegenerate.
(d) Each Tj has a representative Ty.
E T ( M
, &(mK0) 8
which lifts to
f ;
, e r ( M
, e ( m t f + /£)).
This will prove the lemma.
We start the induction at m 1. Since 6(K D) has no base points on M,
&(K J D') has no base points on AT. Then we may choose meromorphic
sections rjr of K J D' on M'r such that (a), (b), and (c) are satisfied for the
germs T* at A of the (a*)~\rjr).
£r(T/r) *s a
meromorphic canonical form on Vr
which is not necessarily holomorphic at;? and at some other subvarieties of Vr. So
there is an effective divisor Gx on Vr, which we also choose to be ample, such that
all of the Tj have representatives rjr G T(M0, 6(K0) ® 0(Gf)). 0(G?) is a quasi-
positive sheaf [16, p. 226]. Let M^ := p^oo) be the polar divisor of pr. Then
M^ M0 is equivalent to the trivial divisor. Then [16, Satz 2.2, p. 273],
H1^^ S(K + G* + Mx- MQr)) = 0. Let ^ = G, + X_1(oo). Then
(4.9) 0 - 0 ( ^ + F? - M0r) - 0 ( ^ + F*) - ©( tf0) ® 6(G?) - 0
is exact since supp(M0 r) n supp(M00) = 0 by construction. The long exact
cohomology sequence for (4.9) proves the m 1 case.
We now assume the existence of the T* for m as in (a)-(d). We must construct
Fm+X and a* G T(M, 0((m + 1)^ - Z)) satisfying (a)-(d) for (m + 1). We first
want a quasipositive sheaf S'on 91Lr such that on 91L (i.e. near E0), 9"is isomorphic
to the subsheaf of QimK^) which is generated by the f-. § will be locally free of
rank 1. To assure quasipositivity, it suffices that ?Thave no base points and that
the (holomorphic) map into projective space given by a basis of r(Afr, 9") be
We have chosen the {T*} SO that (Tf,... ,77*): M ^ P
/ _ 1
is nondegenerate.
Think of pr:
as a nonconstant meromorphic function on 91tr. Then
fh prfj,.. . ,prf/): 91L P 2 / _ 1 is nondegenerate. Let 9" be the subsheaf of
^(mKc^ + F* + M^) generated by the fJr and the pr fy
. We now look at Senear
M''. Let T' in (4.7) be a local generator for 6(mK mJ D') as a subsheaf of
6(mK). In (4.7), T' has a divisor equal to mJ + Df. Let c and i be such that
mJ + Dr = (xcyd). Then
(4.10) T' =
with t(x, j) ^ 0.
T' extends via (4.8) to
(4.11) r = x
m 6 +
( ; c , y)v(x, y)(dx A dy A
with T ET(M0, e(mK0)). Then a local lifting of T to f 6 ^{mK^) will be such
that the divisor of f equals (xma+cymb+d). That is, there is a divisor i/
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