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N ). Conversely, let now f = (ϕ1 , . . , ϕn ) with ϕ1 , . . , ϕn ∈ OX (U) and f (U) ⊂ Y . First of all f is continuous: let Z be any closed subset of Y . Then Z is of the form V (g1 , . . , gm ) for some g1 , . . , gm ∈ A(Y ), and f −1 (Z) = {x ∈ U : gi (ϕ1 (x), . . , ϕn (x)) = 0 for all i = 1, . . , m}. But the functions x → gi (ϕ1 (x), . . , ϕn (x)) are regular on U since plugging in quotients of polynomial functions for the variables of a polynomial gives again a quotient of polynomial functions.

Xn ] we set n V (I) := {x ∈ P : f (x) = 0 for all homogeneous f ∈ I} ⊂ Pn . Obviously, if I is the ideal generated by a set S of homogeneous polynomials then V (I) = V (S). (c) If X ⊂ Pn is any subset we define its ideal to be I(X) := f ∈ K[x0 , . . , xn ] homogeneous : f (x) = 0 for all x ∈ X K[x0 , . . , xn ]. 10 we will denote them by Vp (S) and I p (X), and the affine ones by Va (S) and Ia (X), respectively. 12. (a) As in the affine case, the empty set 0/ = Vp (1) and the whole space Pn = Vp (0) are projective varieties.

4 (PnC is compact in the classical topology). In the case K = C one can give PnC a standard (quotient) topology by declaring a subset U ⊂ Pn to be open if its inverse image under the quotient map π : Cn+1 \{0} → Pn is open in the standard topology. Then PnC is compact: let S = {(x0 , . . , xn ) ∈ Cn+1 : |x0 |2 + · · · + |xn |2 = 1} be the unit sphere in Cn+1 . This is a compact space as it is closed and bounded. Moreover, as every point in Pn can be represented by a unit vector in S, the restricted map π|S : S → Pn is surjective.

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Algebraic Geometry by Andreas Gathmann

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