18 H. B. LAUFER

II

r

to Afr', is a resolution of Vr and that M'r meets the exceptional set in (DTLr

transversely. Af' is the germ of M'r at £0. M0

r

: = p^(0). M0 is the germ of Af0

r

at

We recall [11, p. 143] the canonical extension by 0 from T(M\ 6(mK)) to

T(M0, &(mK))9 where 6(K0) is the canonical sheaf on M0r.

Af

is the exceptional

set in Af'. Let (x, , z) be local coordinates in 911 near (0,0,0) E ^4'. AT = (z =

0}. S, the exceptional set in 911, has reduced equation either xy = 0 or x = 0. For

convenience, we have chosen 911 so that A' n(& — E0) — 0. So p: 911 -* r is

given respectively by t —

xaybzu(x9

y9 z) or t —

xazu(x9

y, z) with a9 b \ and

i/(0,0,0) ¥= 0. The second case is essentially a special case of the first case, so we

shall take / =

xaybzu

: =

xaybzu(x9

y9 z). Recall that M0 = {^ = 0}. Let

Tr

E

T(M\ &(mK)). T' may be given locally by

(4.7) T'=f(x9y)(dxAdy)m.

Extend T' to T on M0 by

(4.8) r =

(xaybu)mf(x,

y)(dx A dy A

dz)m.

Off of £

0

, the vector field ^ =

(\/xaybu)d/dz

satisfies tf(0 = 1 on AT and so

demonstrates the triviality of the normal bundle to M' — A'. Contracting r with 0,

m times, gives T', as needed. On M0, the computed equation for T is independent

of the choices which we have made.

Recall the resolution £: M' - V9 obtained by taking the germ of £r above at A'.

Let 77: Af - Fb e the minimal resolution. There is a blowing down map a: M' -*

M. Let / be the divisor of the Jacobian of the map a. Recall that &(K) denotes

the canonical sheaf on both M and M'. Then

a*: T(M, &(mK)) -» r(M

r

, 6(mK — mJ)) is an isomorphism.

Let D be a positive divisor on A9 the exceptional set in Af. Let D' be the total

transform of D via a'. Then a*: T(M, 0(m# - £)) -^ T(M\ Q{mK - mJ - D'))

is also an isomorphism.

Let D be a cycle on ^ such that D -At 0 for all irreducible components At of

yl and such that K — D has no base points on Af. By Theorem 3.1, it suffices to

take D = -K. Recall that K^ denotes the canonical sheaf on 91tr, and also on

911. We shall show below that, using the above identification a*9 (4.7) and (4.8),

every element of T(Af, Q(mK - D)) lifts to an element of T(M9 © ( m ^ ) ) . (This

was shown for D = 0 by Shepherd-Barron [46, Theorem 12, pp. 14-15].) Since

dim T(M, 6(mK))/T(M9 Q{mK - D)) = x(-™K + D) - x(~™K)9

this will prove the lemma.

For T* E T(Af, 6(mK — 2))), let T' : = a*(r*). Let T denote the canonical

extension of T' by 0 to T(Af0, 6(K0)). By [13], T(M96(rnK - D)) is finitely

generated over T(V9 &v). By induction on m, m ^ 1, we shall find {r*}9 1 j

r

/,