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
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