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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