1.3. Projective varieties 19 Remark 1.2.6. It can be checked that aﬃne varieties with sheaves of reg- ular functions are abstract aﬃne varieties. We claim that, conversely, every abstract aﬃne variety is isomorphic (as a geometric space) to an aﬃne va- riety with its sheaf of regular functions. Indeed, let (X, OX) be an abstract aﬃne variety. Since OX(X) is a finitely generated C-algebra, we can write OX(X) = C[X1,...,Xn]/I for some ideal I. As the elements in OX(X) are C-valued functions on X, OX(X) does not contain nonzero nilpotents hence I is a radical ideal. By the Nullstellensatz (Theorem 1.1.6 c)), we can identify OX(X) with the ring of regular functions C[V(I)] on V(I). Now a morphism of C-algebras from C[V(I)] to C corresponds to a maximal ideal of C[V(I)], hence to a point in V(I). Then, by the property a) of abstract aﬃne varieties we can identify X with V(I) as a set. The Zariski topology on V(I) has the principal open sets as its base, so it now follows from b) that the identification of X and V(I) is a homeomorphism. Finally, by c), OX(Xf) and the ring of regular functions on the principal open set Xf are also identified. This is enough to identify OX(U) with the ring of regular functions on U for any open set U, as regularity is a local condition. The preceding argument shows that the aﬃne variety can be recovered completely from its algebra OX(X) of regular functions, and conversely. Example 1.2.7. In view of Remark 1.2.6, a closed subset of an abstract aﬃne variety is an abstract aﬃne variety (as usual, with the induced sheaf). 1.3. Projective varieties In this section we define projective varieties as subsets of the projective space given by homogeneous polynomial equations. We shall see that a projective variety V has a finite open covering by aﬃne varieties Vi and that the regular functions on V are determined by the regular functions on each Vi. This fact provides the model for the definition of algebraic varieties. The projective n-space over C, denoted by Pn C , or Pn, is the set of equiva- lence classes of (n+1)-tuples (x0, x1,...,xn) of elements in C not all zero un- der the equivalence relation ∼ defined by (x0, x1,...,xn) ∼ (y0, y1,...,yn) ⇔ yi = λxi, 0 ≤ i ≤ n, for some λ ∈ C \ {0}. Equivalently, Pn C is the quo- tient set of An+1\{(0, C . . . , 0)} under the equivalence relation which identifies points lying on the same line through the origin. If V is a finite dimensional C-vector space, we define P(V ) as the quotient of V \{0} by the equivalence relation ∼ defined by v ∼ w ⇔ v = λw for some λ ∈ C \ {0}. An element of Pn C is called a point. We denote by (x0 : x1 : · · · : xn) the equivalence class in Pn C of the element (x0, x1,...,xn) ∈ An+1 C \ {(0,..., 0)}. We call (x0 : x1 : · · · : xn) the homogeneous coordinates of the point P . They are defined up to a nonzero common factor in C.

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