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.
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