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