8 1. Warming up to Enumerative Geometry

Deﬁnition 1.4. The projectivization P(V ) of V is the quotient of

V − 0 by the equivalence relation x ∼ x if x = λx for some λ ∈

C∗.

In the statement of Deﬁnition 1.4, the zero element of V has been

denoted by 0.

See Exercise 3. Another description of P(V ) is given in Exercise 5.

Let’s return to our ﬁrst enumerative question about two lines in

the plane. We replace the plane by C2, which we then compactify by

enlarging it to P2. More generally, Pn has a subset

U0 = {(x0, . . . , xn) ∈

Pn

| x0 = 0}

which is in one-to-one correspondence with

Cn

via the map

φ0 : U0 →

Cn

: (x0, . . . , xn) →

x1

x0

,

x2

x0

, . . . ,

xn

x0

.

The inverse map is given by

ψ0 :

Cn

→ U0, (x1, . . . , xn) → (1, x1 . . . , xn).

So

Pn

is an enlargement of

Cn

(in fact, it’s a compactiﬁcation). The

complement of U0 in

Pn

is naturally in one-to-one correspondence

with

Pn−1

(Exercise 6).

Similarly, we have subsets Ui deﬁned by xi = 0 for each 0 ≤ i ≤ n.

Each of these Ui is in one-to-one correspondence with

Cn,

as will be

seen explicitly in Example 4.20.

We have a notion of homogeneous polynomials on Pn: these are

the polynomials for which all terms have the same total degree. If

F (x0, . . . , xn) is homogeneous, then its zero locus

Z(F ) = {x ∈

Pn

| F (x0, . . . , xn) = 0}

is well deﬁned and is called the hypersurface deﬁned by F . See Exer-

cise 7. We refer to F as a deﬁning equation of Z(F ).

Given a hypersurface Z ⊂

Pn,

its deﬁning equation is far from

unique; e.g. Z(F ) = Z(λF ) for any λ ∈

C∗,

and Z(F ) = Z(F

n)

for any positive integer n. The degree of Z is the minimal degree

of a deﬁning equation for Z. Taking the minimal degree eﬀectively

eliminates the possibility of introducing extraneous powers in F or

Deﬁnition 1.4. The projectivization P(V ) of V is the quotient of

V − 0 by the equivalence relation x ∼ x if x = λx for some λ ∈

C∗.

In the statement of Deﬁnition 1.4, the zero element of V has been

denoted by 0.

See Exercise 3. Another description of P(V ) is given in Exercise 5.

Let’s return to our ﬁrst enumerative question about two lines in

the plane. We replace the plane by C2, which we then compactify by

enlarging it to P2. More generally, Pn has a subset

U0 = {(x0, . . . , xn) ∈

Pn

| x0 = 0}

which is in one-to-one correspondence with

Cn

via the map

φ0 : U0 →

Cn

: (x0, . . . , xn) →

x1

x0

,

x2

x0

, . . . ,

xn

x0

.

The inverse map is given by

ψ0 :

Cn

→ U0, (x1, . . . , xn) → (1, x1 . . . , xn).

So

Pn

is an enlargement of

Cn

(in fact, it’s a compactiﬁcation). The

complement of U0 in

Pn

is naturally in one-to-one correspondence

with

Pn−1

(Exercise 6).

Similarly, we have subsets Ui deﬁned by xi = 0 for each 0 ≤ i ≤ n.

Each of these Ui is in one-to-one correspondence with

Cn,

as will be

seen explicitly in Example 4.20.

We have a notion of homogeneous polynomials on Pn: these are

the polynomials for which all terms have the same total degree. If

F (x0, . . . , xn) is homogeneous, then its zero locus

Z(F ) = {x ∈

Pn

| F (x0, . . . , xn) = 0}

is well deﬁned and is called the hypersurface deﬁned by F . See Exer-

cise 7. We refer to F as a deﬁning equation of Z(F ).

Given a hypersurface Z ⊂

Pn,

its deﬁning equation is far from

unique; e.g. Z(F ) = Z(λF ) for any λ ∈

C∗,

and Z(F ) = Z(F

n)

for any positive integer n. The degree of Z is the minimal degree

of a deﬁning equation for Z. Taking the minimal degree eﬀectively

eliminates the possibility of introducing extraneous powers in F or