1. Warming up to Enumerative Geometry 5 (0, 1) as a point of P1. So P1 is obtained from a copy of C by adding a single point. This point can be thought of as the point at infinity. To see this, consider a complex number t, and identify it with a point of U0 using ψ0 i.e., we identify it with ψ0(t) = (1, t). Now let t → ∞. The beautiful feature is that the limit now exists in P1! To see this, rewrite (1, t) as (1/t, 1) using scalar multiplication by 1/t. This clearly approaches (0, 1) as t → ∞, so (0, 1) really should be thought of as the point at infinity! We have been deliberately vague about the precise meaning of limits in P1. This is a notion from topology, which we will deal with later in Chapter 4. The property that limits exist in a topological space is a consequence of the compactness of the space, and the pro- cess of enlarging C to the compact space P1 is our first example of the important process of compactification. This makes the solutions to enumerative problems well-defined, by preventing solutions from going off to infinity. A precise definition of compactness will be given in Chapter 4. We now have to modify our description of complex polynomials by associating to them polynomials F (x0, x1) on P1. Before turning to their definition, note that the equation F (x0, x1) = 0 need not make sense as a well-defined equation on P1, since it is conceivable that a point could have different representatives (x0, x1) and (x 0 , x 1 ) such that F (x0, x1) = 0 while F (x 0 , x 1 ) = 0. We avoid this problem by requiring that F (x0, x1) be a homogeneous polynomial i.e., all terms in F have the same total degree, which is called the degree of F . So (2) F (x0, x1) = d i=0 aix0x1−idi is the general form of a homogeneous polynomial of degree d. If λ ∈ C∗, then we compute F (λx0, λx1) = λdF (x0, x1), so F (x0, x1) = 0 if and only if F (λx0, λx1) = 0, and the equation F (x0, x1) = 0 is a well-defined condition on a point (x0, x1) ∈ P1. Having enlarged C to P1, we correspondingly need to “extend” an arbitrary polynomial f(x) to a homogeneous polynomial F (x0, x1).

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