WEAK SIMULTANEOUS RESOLUTION
Lx. Since IT X maps Stein covers to Stein covers, there is an exact sequence
(2.1) o - r(x, e(Lx -
)) - r(c, 6(L)) - r(c,ft)
- ff *( *, 0(Z^ - c)) - Hl(C9 0(L)) - 0.
(2.1) corresponds to [24, (1.11), p. 116].
Now suppose that C is an irreducible curve. Let g' be the arithmetic genus of
C, i.e., g' = dim H\C, 0C). Let g be the geometric genus of C, i.e. the genus of X.
Then g' = g + 8. Let c(L) be the Chern class of L. Then c(L^) = c(L). From
(2.1), it follows that for c(L) 2g' - 2,
(2.2) dim T(C, 0(L)) = c(L) - 1 - g ' and /T(C, 0(L)) = 0.
Recall [45, Proposition IV, p. 75] that if C is an irreducible subvariety of the
two-manifold M and if K is the canonical bundle on M,
(2.3) g' = 1 + H C - C + C-A-).
We now wish to re-examine the proof of [27, Theorem 3.2, p. 603; 23, p. 246]
for the nongood resolution case. Let M be a strictly pseudoconvex two-manifold
with connected exceptional set A Let A — \JAt be the decomposition of A into
irreducible components. Let L be a line bundle over M with 0(L) its sheaf of
germs of sections. Recall that LAt is just the Chern class of L restricted to At.
Suppose that LAj KAt for all /. Then [23, p. 246], H\M9 0(L)) = 0.
In proving this, we may proceed as follows. Let Z0 = 0, Zx — Ai,... 9Zk =
At + Zk_x,.. ,9Zi=Z = Ai+ Z7_! be a computation sequence for Z, the
fundamental cycle [27, Proposition 4.1, p. 607; 29, p. 1259]. ,4,-Z^O , all /;
At - Zk_ j 0 for 1 k I; Ai • Z0 = 0, of course.
Now consider the computation sequence in reverse order: Let Ah — Ai9... ,Aj
= At with
= l - k+ 1,... ,AJt = Ai}9 K k I. Let 70 = 0 = Z - Z/9 7, =
= Z - Z
, . . . , 7 , = 7 , _
+ ^ = Z - Z
_ „ . . . , 7
= Z - Z
= Z. Look
at (L — Yk_x)-Ajk. Let g/ denote the arithmetic genus of At. Let r — I — k + 1.
Recall [45, Proposition IV. 5, p. 75] that Ar K = 2g/ - 2 - ^ - ^ Then
(2.4) (L - 7 ^ , ) - ^ = (L - Z + Z
=(L-Z + Zr_x + ^ ) - ^
2 g / - 2 - ^
, Z + ^
For r 1, which is equivalent to k /, we have v ^ - Z ^ ^ O . So
(L - Yk_x)-Ajk 2g'ik - 2. For r = 1, or k = /, we choose ^ so that ^ • Z 0.
Such an Aix exists since Z - Z 0. Then also (L — Yt_x)-Aj 2gj — 2. Recall
that for a divisor D 0, 6(L — D) denotes those germs of sections of L which
vanish to the order given by D. By (2.2), for all 1 k /, the line bundle
L — Yk_x has trivial cohomology on Aj . Upon reaching L — Z = L — Yh we
may replace L by L — Z and repeat the argument. So, as in the proof of [27,
Theorem 3.2, p. 603], which is essentially [14, §4 Satz 1, p. 355] we have proved
2.1. Let M be a strictly pseudoconvex two-manifold with connected
exceptional set A = \JAt. Let L be a line bundle on M such that LAi^KAi for