1.3. Projective varieties 21 We define the Zariski topology in Pn C as the topology having the projec- tive varieties as closed sets. Once we have a topological space, the notion of irreducible subset applies as in the affine case. The projective space Pn with the Zariski topology is a Noetherian space hence we can define the dimension of a subset of Pn also as in the affine case. Example 1.3.6. If f is a non constant homogeneous polynomial in the ring C[X0,X1,...,Xn], V(f) is a hypersurface in Pn C . If f is a linear homoge- neous polynomial, V(f) is a hyperplane. Let us consider the map ι : An Pn given by (x1, . . . , xn) (1 : x1 : · · · : xn). It is injective and its image is Pn \H, for H := {(x0 : x1 : · · · : xn) : x0 = 0}. The hyperplane H is called hyperplane at infinity. Example 1.3.7. If R = ⊕d≥0Rd is a graded ring, ⊕d0Rd is a homogeneous ideal of R. In particular, the ideal I0 of C[X0,X1,...,Xn] generated by X0,X1,...,Xn is a proper radical homogeneous ideal of C[X0,X1,...,Xn]. We have V(I0) = ∅. This ideal I0 is sometimes called the irrelevant ideal as it does not appear in the 1-1 correspondence between projective varieties in Pn C and radical ideals of C[X0,X1,...,Xn]. (See Proposition 1.3.8 below.) To each subset S Pn C we associate the homogeneous ideal I(S) gener- ated by {f C[X0,X1,...,Xn] : f is homogeneous and f(P ) = 0 for all P S}. As in the affine case, ideals of the form I(S) are radical ideals. If Y is a projective variety, we define the homogeneous coordinate ring of Y to be C[Y ] = C[X0,X1,...,Xn]/I(Y ). We now state the projective Nullstellensatz, which has a similar formu- lation to the affine one, except for the adjustment due to the fact that the ideal I0 defined in Example 1.3.7 has no common zeros. It can be easily de- duced from the affine Nullstellensatz. The statement on irreducible varieties applies here as well. Proposition 1.3.8. The mappings V and I set a bijective correspondence between the closed subsets of Pn C and the homogeneous radical ideals of C[X0,X1, . . . , Xn] other than the irrelevant ideal I0. Irreducible projective varieties correspond to homogeneous prime ideals un- der this correspondence.
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