1. Warming up to Enumerative Geometry 5
(0, 1) as a point of
P1.
So
P1
is obtained from a copy of C by adding
a single point.
This point can be thought of as the point at infinity. To see
this, consider a complex number t, and identify it with a point of U0
using ψ0; i.e., we identify it with ψ0(t) = (1, t). Now let t ∞.
The beautiful feature is that the limit now exists in
P1!
To see this,
rewrite (1, t) as (1/t, 1) using scalar multiplication by 1/t. This clearly
approaches (0, 1) as t ∞, so (0, 1) really should be thought of as
the point at infinity!
We have been deliberately vague about the precise meaning of
limits in P1. This is a notion from topology, which we will deal with
later in Chapter 4. The property that limits exist in a topological
space is a consequence of the compactness of the space, and the pro-
cess of enlarging C to the compact space P1 is our first example of
the important process of compactification. This makes the solutions
to enumerative problems well-defined, by preventing solutions from
going off to infinity. A precise definition of compactness will be given
in Chapter 4.
We now have to modify our description of complex polynomials
by associating to them polynomials F (x0, x1) on P1. Before turning
to their definition, note that the equation F (x0, x1) = 0 need not
make sense as a well-defined equation on P1, since it is conceivable
that a point could have different representatives (x0, x1) and (x0, x1)
such that F (x0, x1) = 0 while F (x0, x1) = 0. We avoid this problem
by requiring that F (x0, x1) be a homogeneous polynomial; i.e., all
terms in F have the same total degree, which is called the degree of
F . So
(2) F (x0, x1) =
d
i=0
aix0x1−idi
is the general form of a homogeneous polynomial of degree d. If λ
C∗,
then we compute F (λx0, λx1) =
λdF
(x0, x1), so F (x0, x1) = 0
if and only if F (λx0, λx1) = 0, and the equation F (x0, x1) = 0 is a
well-defined condition on a point (x0, x1)
P1.
Having enlarged C to
P1,
we correspondingly need to “extend”
an arbitrary polynomial f(x) to a homogeneous polynomial F (x0, x1).
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