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We can think of this map as sending the complex line x = 1 onto the complex curve
with t =+1, -1 mapping onto the origin. The map is onto and one-to-one except for
these two points. The line x = 1 is obviously a curve with no singularities. Later on
we shall look at this as an example of how singularities are “resolved.” Although c is
not globally one-to-one, it is locally so. Small enough neighborhoods of +1 get mapped
in a one-to-one fashion onto a set in
C
, similarly for a small neighborhood of -1. See
Figure 10.15 again, where the points
p
,
q
,
r
, and
s
get mapped onto
p
¢,
q
¢,
r
¢, and
s
¢,
respectively. One can think of this as having broken the curve
C
into more primitive
parts, in analogy with how we factored polynomials into their irreducible factors and
expressed varieties as unions of irreducible varieties. In our case the polynomial
(
)
=-
2
2
1
(
)
fXY
,
Y
X
-
X
in the ring
C
[X,Y] that defines
C
is irreducible. But then again, we are now dealing
with a local decomposition of the curve, not a global one as in the case of factoring
polynomials. Is there another ring in which f factors? Factoring into polynomials did
not work. Factoring into rational functions would also not work. However, suppose
we consider holomorphic functions in two variables in a neighborhood
U
of the origin.
In this case we
can
factor f as f = f
1
f
2
, where
(
)
=+
fxy
,
y x
1
1
-
x
,
1
(
)
=-
fxy y x
,
-
x
,
2
and we have used one of the branches of the square root function. To avoid
the problem with the neighborhood
U
not being well defined, one passes to equiva-
lence classes of holomorphic functions. In other words, we have a factorization in
the ring of holomorphic branches at the origin, where “branch” is being used in the
sense described above. We have found a ring of the type we were looking for. Singu-
larities of curves can be detected by checking the “irreducibility” of the curve in this
ring.
We are done with the introductory remarks for this section. See [BriK81] for a
much more complete discussion of how complex analysis comes into the picture for
algebraic geometry. Being part of analysis, the methods and approaches described
above make heavy use of convergence and differentiability, concepts intimately con-
nected with the topology of the complex plane. Topology also plays a role in algebraic
geometry, but algebraic geometers prefer not to have to worry about convergence of
series and approach the same questions in a purely algebraic way. They are guided
by what is known from analysis, but replace convergent series with formal power
series where convergence is not an issue. Hopefully, this abstract algebraic approach,
which we shall now describe, will make more sense with our background discussion.
We could start with the most concrete case of an affine plane curve defined by an
equation
(
)
= 0
fXY
,
.
The plan would then be to express X and Y as formal power series in a variable t. This
is plausible because an implicit function type theorem basically states that this is pos-
