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 xd, 1 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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