8
H. B. LAUFER
THEOREM
3.3. Let V and Sw be defined as above. Let j: X - V be the monoidal
transformation with respect to Sm. Let X denote the normalization of X. Let
a: X - V be the induced map. For m 2, a: X - V is the RDP resolution of V.
For m 3, X is already normal and J: X - V is the RDP resolution of V.
Using the same example as in [46, p. 8] we can see that the conclusions of
Theorem 3.3 do not always hold for m 1. Also, for m 2, X need not be
normal. Thus, consider the singularity/' = (0,0,0) in V {(x, y, z)\z2 x3 +
y6}. p is a simple elliptic singularity [44]. Let ir\ M - Vbe the minimal resolution.
Let A = ir~\p) be the exceptional set. Then A is an elliptic curve with A A = - 1 .
Sj is isomorphic to the sheaf of ideals generated by (x, y, z). Blowing up at S,
and then normalizing yields [31, Lemma 2.2, p. 317] a space with a simple elliptic
singularity equivalent to (0,0,0) in {(x, y9
z)\z2

y(x3
+
y3)}.
S2
*s
isomorphic
to the sheaf of ideals generated by (x,
y29
z). Blowing up at S2 yields a
nonnormal space.
Theorem 3.3 is really a local theorem, as may be seen as follows. Let co: V - V
be the normalization of V9 a local construction in all dimensions. Since V is of
dimension two, it has only isolated singularities. Hence resolving V (and thereby
resolving V) is a local construction. Hence §m is defined locally, and so j: X -* V
is defined locally. Thus in proving Theorem 3.3, we may assume that Fis Stein.
Suppose that V is reducible at y E V. Let V U ^ be the decomposition of V
into irreducible components. Then in resolving V, we may separately resolve each
VJ9 Try. Mj - Vj. Let SmJ denote irj{Q{mK)). Then §my9 the stalk of Sm at y, has a
natural direct sum decomposition Sm ^ ~ ©§m,y,y- Let fy: ^ - Vj be the mon-
oidal transformation of V- with respect to §.. Then Jfis isomorphic to the disjoint
union of the X-. So in proving Theorem 3.3, we may assume that V is irreducible
dXy.
Let co: V Fb e the normalization of K. Then
co-1^)
is just one point, y, since
V may be assumed to be irreducible at y. Suppose that co resolves V, i.e. V is
smooth aty. 6(mK) is isomorphic to Seneary. Then Sw is isomorphic to 0K, the
sheaf of germs of weakly holomorphic functions. Let zx and z2 be local coordi-
nates near y. Then zx and z2 are also weakly holomorphic functions near y. So
generators for Sm include 1, z, and z2. Then X ^ V.
Recall [41] the definitions of the various pull-backs of a sheaf. Let \p: N - Fb e
a proper modification. Let S be a coherent sheaf on V. ^*S has stalk a t i G i V
given by (^*S)^ : = S^(JC) ® 0 ^ , where 0^
x
is given the structure of an &VMx)
module via ^ and ® is taken over 0F^(jc). Let T(\f/*$) be the torsion subsheaf of
i//*S. Then S ° ^ := i//*S/!T(i//*S).
PROPOSITION 3.4. Ler \p: N -^ V be a proper modification of the two-dimensional
space V. Suppose that N is normal. Let r: M N be a resolution of N. Let
7T = \p o r. Suppose that IT: M V is the minimal resolution. Recall Sm = irJd(mK)
from above. Let m 2. 77ze« Sm o 77 = 0(mAT). Ifr^(&(mK)) is locally free, then
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