1. Warming up to Enumerative Geometry 7

Proof. By induction on d. The case d = 1 is already proven. In

the general case, factor out all powers of x0, writing F (x0, x1) =

x0G(x0,

r

x1) where G is homogeneous of degree d−r and is not divisible

by x0. Then the point (0, 1) at inﬁnity is a root of F with multiplicity

r. If r 0, then G has d − r roots by induction. These roots are also

roots of F , and when they are combined with the r roots at (0, 1), we

have found all d roots and we are done in this case.

So we may assume that F is not divisible by x0. This implies that

it has a nonzero term involving x1, d so that its dehomogenization f(x)

also has degree d. Now the fundamental theorem of algebra implies

that f(x) has a complex root x = a. Write f(x) = (x − a)h(x) where

h has degree d − 1. Then F (x0, x1) = (x1 − ax0)H(x0, x1), where H,

the homogenization of h, has degree d − 1. By induction, H has d − 1

roots. These are all roots of F as well, and F has the additional root

(1, a), giving d roots in all.

An essential role was played by the fundamental theorem of al-

gebra, which is why we had to use complex coeﬃcients. The proof

shows that a similar result would have held if we had used coeﬃcients

in an arbitrary algebraically closed ﬁeld.

Later, in Chapter 11, we will rederive Theorem 1.2 using elemen-

tary instances of deep ideas from physics including supersymmetry

and topological quantum ﬁeld theories.

The notion of

P1

generalizes.

Deﬁnition 1.3. The n-dimensional complex projective space CPn

(or just Pn) is the set of all ordered (n+1)-tuples of complex numbers

{x = (x0, . . . , xn) ∈

Cn+1

| x = (0, . . . , 0)}

where we identify pairs which are scalar multiples of each other: x =

λx for some λ ∈

C∗.

Complex projective space plays a fundamental role in complex

algebraic geometry.

The process of projectivization can be generalized to any complex

vector space V , including inﬁnite-dimensional spaces.

Proof. By induction on d. The case d = 1 is already proven. In

the general case, factor out all powers of x0, writing F (x0, x1) =

x0G(x0,

r

x1) where G is homogeneous of degree d−r and is not divisible

by x0. Then the point (0, 1) at inﬁnity is a root of F with multiplicity

r. If r 0, then G has d − r roots by induction. These roots are also

roots of F , and when they are combined with the r roots at (0, 1), we

have found all d roots and we are done in this case.

So we may assume that F is not divisible by x0. This implies that

it has a nonzero term involving x1, d so that its dehomogenization f(x)

also has degree d. Now the fundamental theorem of algebra implies

that f(x) has a complex root x = a. Write f(x) = (x − a)h(x) where

h has degree d − 1. Then F (x0, x1) = (x1 − ax0)H(x0, x1), where H,

the homogenization of h, has degree d − 1. By induction, H has d − 1

roots. These are all roots of F as well, and F has the additional root

(1, a), giving d roots in all.

An essential role was played by the fundamental theorem of al-

gebra, which is why we had to use complex coeﬃcients. The proof

shows that a similar result would have held if we had used coeﬃcients

in an arbitrary algebraically closed ﬁeld.

Later, in Chapter 11, we will rederive Theorem 1.2 using elemen-

tary instances of deep ideas from physics including supersymmetry

and topological quantum ﬁeld theories.

The notion of

P1

generalizes.

Deﬁnition 1.3. The n-dimensional complex projective space CPn

(or just Pn) is the set of all ordered (n+1)-tuples of complex numbers

{x = (x0, . . . , xn) ∈

Cn+1

| x = (0, . . . , 0)}

where we identify pairs which are scalar multiples of each other: x =

λx for some λ ∈

C∗.

Complex projective space plays a fundamental role in complex

algebraic geometry.

The process of projectivization can be generalized to any complex

vector space V , including inﬁnite-dimensional spaces.