6 1. Warming up to Enumerative Geometry Letting d be the degree of f(x), we define its homogenization as (3) F (x0, x1) = xdf 0 x1 x0 . Note that the homogenization satisfies (4) f(x) = F (ψ0(x)) = F (1, x). But notice that (4) does not uniquely specify F (x0, x1) given f(x), since multiplication of F by any power of x0 will not alter the va- lidity of (4). In other words, we have homogeneous polynomials F of different degrees which satisfy (4). So we can, if we want to, replace the occurrence of d in (3) by an integer e d and put F (x0, x1) = x0f(x1/x0), e which is easily checked to be the unique homogeneous solution to (4) of degree e. However, when we speak of the homogenization of f(x), our meaning is (3), i.e., the choice e = d. The process of going from F (x0, x1) to f(x) by (4) is referred to as dehomogenization.2 Now let’s go back to the case of a polynomial f(x) of degree 1. The homogenization of f(x) = a0x + a1 as a polynomial of degree 1 is F (x0, x1) = a0x1 + a1x0. Note how the case a0 = 0 makes no difference and is treated on equal footing with the case a0 = 0: as long as f(x) is not the zero polynomial (equivalently, F is not the zero polynomial), there is exactly one root in P1, namely (x0, x1) = (−a0, a1). The homogenization of a general degree 2 polynomial is given by F (x0, x1) = a0x1 2 + a1x0x1 + a2x0. 2 Again, as long as F is not the zero polynomial, there are exactly two roots in P1 (including multiplicity). For example, if a0 = 0, then the roots are (0, 1) and (−a1, a2). Note that this is exactly what we found earlier: identifying C with U0 via (1), then the point (0, 1) is thought of as being at infinity, and the point (−a1, a2) is identified with the point φ0((−a1, a2)) = −a2/a1 found earlier. See Exercise 4. We can now state an easy generalization: Theorem 1.2. Any nonzero homogeneous polynomial F (x0, x1) of degree d has exactly d roots in P1 including multiplicity. 2 We could more precisely have referred to this as dehomogenization with respect to x0 to emphasize that x0 is the variable that gets set to 1.
Previous Page Next Page