The name is absent

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 = R^p × 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, R^p,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 V_u of и in T.

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

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

(a) for each geW_f, G(U_x) ⊂ V_u ;

(b) for each he W_j-, there exists a geW_j∙ arbitrarily close to h
such that GI U_x 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.

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