22 1. Affine and Projective Varieties We shall now see that the projective space Pn has an open covering by affine n-spaces. Let Hi be the hyperplane {(x0 : x1 : · · · : xn) : xi = 0} and let Ui be the open set Pn \ Hi. Then Pn is covered by the Ui, 0 i n because if P = (x0 : x1 : · · · : xn) Pn at least one of the coordinates xi is not zero, hence P Ui. We define a mapping ϕi : Ui An (x0 : x1 : · · · : xn) x 0 xi , . . . , x i−1 xi , x i+1 xi , . . . , xn xi . It is well defined as the ratios xj/xi are independent of the choice of homo- geneous coordinates. Proposition 1.3.9. The map ϕi is a homeomorphism of Ui with its induced topology to An with its Zariski topology. Proof. The map ϕi is clearly bijective so it will be sufficient to prove that the closed sets of Ui are identified with the closed sets of An by ϕi. We may assume i = 0 and write U for U0 and ϕ for ϕ0. Let R = C[X0,X1,...,Xn],A = C[Y1,...,Yn]. We define a map α from the set Rh of homogeneous polynomials in R to A and a map β from A to Rh as fol- lows. For f Rh, put α(f) = f(1,Y1,...,Yn) for g A of total degree e, the polynomial Xe 0 g(X1/X0,...,Xn/X0) is a homogeneous polynomial of degree e which we take as β(g). From the definitions of α and β, we easily obtain the equalities ϕ(V(S)) = V(α(S)), for a subset S of Rh, and ϕ−1(V(T )) = V(β(T )), for a subset T of A. Hence both ϕ and ϕ−1 are closed maps, so ϕ is a homeomorphism. Corollary 1.3.10. A subset S of Pn is closed if and only if its intersections S Ui are all closed (Ui being identified with An via the mapping ϕi defined above). If Y is a projective variety, then Y is covered by the open sets Y Ui, 0 i n, which are homeomorphic to affine varieties via ϕi. Remark 1.3.11. For a homogeneous polynomial f in R = C[X0,X1,...,Xn], the polynomial α(f) = f(1,Y1,...,Yn) A = C[Y1,...,Yn] is called the de- homogenization of f with respect to the variable X0. For a polynomial g A of total degree e, the polynomial β(g) = Xe 0 g(X1/X0,...,Xn/X0) is called the homogenization of g with respect to the new variable X0. We can define analogously dehomogenization and homogenization with respect to the other variables. We can easily see α(β(g))) = g for any polynomial g in A. On the contrary, β α is not the identity on Rd. For example, β(α(X0)) = 1. From the proof of Proposition 1.3.9, it follows that if Y is a projective variety, the ideal corresponding to Y U0, as an affine variety of U0 identified with An, is {α(f)|f I(Y )}.
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