(0, 1) as a point of
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
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) =
is the general form of a homogeneous polynomial of degree d. If λ ∈
then we compute F (λx0, λx1) =
(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) ∈
Having enlarged C to
we correspondingly need to “extend”
an arbitrary polynomial f(x) to a homogeneous polynomial F (x0, x1).