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58


RICE UNIVERSITY STUDIES


where β ≥ O and β ≠. In this case p is a non degenerate exceptional
point.

J2(2,2) can be identified with ten-dimensional complex euclidean space.
Let
p be an exceptional point of a manifold M2 embedded in C2 in such
a way that
y = 0. Since the condition y = 0 is not invariant under coor-
dinate changes on
M2, the submanifold S2 °f J2(2,2) which arises from
the fact that
p is an exceptional point with y = 0 is not a singularity mani-
fold. However, if
M2 has only a finite number of exceptional points under
an embedding
f into C2, we may apply the following lemma at each of
these points.

We have the notation:

T = Rp × J'(n,p)

F: R"→T-.x→(f(x),J'(f)(x))

C: R"->T1X→(∕(x),Z(g)(x)).

Lemma 3.1 (Local lemma). Suppose f L(Rn, Rp,s) and N T is an
(s—r) differentiable regular submanifold of codimension q. If (s—r) >
max (n-q,0), then for each xeR" and each ueN<=T such that
/(χ) = и we can find".

(1) A neighborhood Vu of и in T.

(2) A neighborhood Wj of f in L(R", Rp,s).

(3) A compact neighborhood Ux of x in R" such that

(a) for each geWf, G(Ux) Vu ;

(b) for each he Wj-, there exists a geWj∙ arbitrarily close to h
such that
GI Ux is transversal to N.

If we set N = C2 × S2 and note that the real codimension of S2 in
J2(2,2) is 4, we find by applying the lemma at each exceptional point,
that arbitrarily close to the embedding
f is an embedding g which has a
finite number of exceptional points with y ≠ 0 at each such point.

We use the lemma again with the condition y = 0 replaced by the con-
dition
I β I = -2-1 y I. Thus the set of embeddings under which a manifold
has no exceptional points or a finite number of exceptional points with
y ≠ 0 and
I β I ≠ -2∙ I y I at each such point ɪs dense in the set of all embeddings.
Therefore the non degenerate embeddings of
M2 into C2 are an open dense
set in the set of all ⅛
00 embeddings.

4. The Gauss Mapping and Intersection Theory

Let M ⊂ C2 be a compact oriented two-manifold with a given orienta-
tion. Assume
M has been embedded in C2 by a non degenerate embedding.



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