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20 1. Affine and Projective Varieties Example 1.3.1. We can consider a map A1 → P1 given by x → (1 : x). It is injective and the image contains all points in P1 except (0 : 1). This ”extra point” is called point at infinity. If we consider in the affine plane a line l and a point P not in l, then every line through P cuts l in a point, except the parallel line to l. Adding to l an extra point at infinity, we obtain a 1-to-1 correspondence between the set of lines through P and the set of points in l. We will now define projective varieties in Pn C in a way analogous to the definition of affine varieties. Due to the nonuniqueness of homogeneous coordinates, the fact that a polynomial in C[X0,X1,...,Xn] vanishes at a point of Pn C is not well determined for an arbitrary polynomial. We need to introduce homogeneous polynomials and homogeneous ideals of the ring C[X0,X1,...,Xn]. We shall define more generally the notion of a graded ring. Definition 1.3.2. A graded ring is a commutative ring R together with a decomposition R = ⊕d≥0Rd of R into the direct sum of subgroups Rd such that for any d, e ≥ 0, Rd.Re ⊂ Rd+e. An element of Rd is called a homogeneous element of degree d. Thus any element of R can be written uniquely as a finite sum of homogeneous elements. Example 1.3.3. The ring of polynomials R = C[X0,X1,...,Xn] can be made into a graded ring by taking Rd to be the subgroup containing all linear combinations of monomials of total degree d in X0,X1,...,Xn. If f ∈ Rd, λ ∈ C, we have f(λ x0,λ x1,...,λ xn) = λd f(x0,x1,...,xn) hence the fact that (x0 : x1 : · · · : xn) is a zero of f is well determined. Definition 1.3.4. If R = ⊕d≥0Rd is a graded ring, an ideal I ⊂ R is a homogeneous ideal if I = ⊕d≥0(I ∩ Rd), i.e. if all homogeneous parts of every element of I also belong to I. We define a projective variety as the set of common zeros in Pn C of a finite collection of homogeneous polynomials in C[X0,X1,...,Xn]. To each homogeneous ideal I of C[X0,X1,...,Xn] we associate the set V(I) of its common zeros in Pn C . Taking into account that C[X0,X1,...,Xn] is a Noe- therian ring and Exercise 13, each ideal of C[X0,X1,...,Xn] has a finite set of homogeneous generators. Therefore the set V(I) is a projective variety. Proposition 1.3.5. The correspondence V satisfies the following equalities: a) Pn C = V(0), ∅ = V(C[X0,X1,...,Xn]), b) If I and J are two homogeneous ideals of C[X0,X1,...,Xn], V(I) ∪ V(J) = V(I ∩ J), c) If Iα is an arbitrary collection of homogeneous ideals of C[X0,X1,...,Xn], ∩αV(Iα) = V( ∑ α Iα).