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RICE UNIVERSITY STUDIES
where β ≥ O and β ≠ ⅛. 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 ⅛00 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.