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).

(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).