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Projective Varieties Chapter 1 Projective Varieties 1.1 Projective Space and Algebraic Sets 1.1.1 Definition. Consider An+1 = An+1(|). The set of all lines in An+1 passing through the origin 0 = (0;:::; 0) is called the n-dimensional projective space and is denoted by Pn(|), or simply Pn when | is understood. We also have the identification n n+1 ∗ P = (A n f0g)=| ; ∗ where (x1; : : : ; xn+1) ∼ (λx1; : : : ; λxn+1) for all λ 2 | , i.e. , two points in An+1nf0g are equivalent if they are on the same line through the origin. An ele- n ment of P is called a point. If P is a point, then any (n+1)-tuple (a1; : : : ; an+1) in the equivalence class P is called a set of homogeneous coordinates for P . Equivalence classes are often denoted by P = [a1 : ··· : an+1] to distinguish from the affine coordinates. Note that [a1 : ··· : an+1] = [λa1 : ··· : λan+1] for all λ 2 |∗. We defined Pn as the collection of all one dimensional subspaces of the vector space An+1, but Pn may also be thought of as n+1 (overlapping) copies of affine n-space. Indeed, we can express any any point [x1 : ··· : xn1 ] 2 Ui in terms of n affine coordinates: x1 xn+1 [x1 : ··· : xn+1] = : ··· : 1 : ··· : : xi xi ∼ n Thus, Ui = A . 1.1.2 Example. Consider P2, where [x : y : z] are the homogeneous coordi- nates. Then nh y z i o U = x : : 2 2 j x 6= 0 x x x P =∼ f[1 : u : v] j u; v 2 |g = f(u; v) j u; v 2 |g 2 = A ; 1 so (u; v) are the affine coordinates on Ux. Similarly, x z U = f[x : y : z] 2 2 j y 6= 0g = : 1 : 2 2 j y 6= 0 ; y P y y P and nhx y i o U = f[x : y : z] 2 2 j z 6= 0g = : : 1 2 2 j y 6= 0 : z P z z P Moreover, An can be considered as a subspace of Pn, where the inclusion n n is given by identifying A with Un+1 ⊆ P , i.e. the inclusion is the map ' : n n A ! P given by '(u1; : : : ; un) = [u1 : ··· : un : 1]. One could also introduce a 1 to any other position, but we will usually use this convention. n For each i = 1; : : : ; n + 1, Hi = f[x1 : ··· : xn+1] j xi = 0g = P n Ui is a hyperplane, which can be identified with Pn−1 by the correspondence [x1 : ··· : 0 : ··· : xn+1] ! [x1 : ··· : xi−1 : xi+1 : ··· : xn+1]: In particular, Hn+1 is often denoted H1 and is called the hyperplane at infinity, and n ∼ n n−1 P = Un+1 [ H1 = A [ P ; n n so P is the union of a copy of A and H1, which can be seen as the set of all directions in An. 1.1.3 Examples. (i) P0 = f1g is a single point. (ii) P1 = A1 [ P0 = A1 [ f1g, the one point compactification of A1. (iii) P2 = A2 [ P1, where the copy of P1 here is often referred to as the line at infinity, and is denoted by `1. Remarks. In Pn, any two lines intersect. For example, consider two distinct parallel lines in A2: L : au + bv + c = 0 L0 : au + bv + c0 = 0; where c 6= c0. If one considers A2 as the subset 2 2 f[u : v : 1] 2 P j u; v 2 |g = f[x : y : z] 2 P j x; y 2 |; z 6= 0g = Uz of P2, we can see that the equations can be rewritten as follows by letting u = x=z and v = y=z: ax + by + cz = 0 ax + by + c0z = 0: Subtracting the two equations, we get z(c − c0) = 0, so z = 0, as we assumed that c 6= c0. Hence any solution is of the form [x : y : 0] and lies on the line 2 at infinity. Substituing this value of z back into either equation gives that ax + by = 0, so x = bt and y = −at for some t 2 |. Therefore, the solutions to the system are ∗ f(bt; −at; 0) j t 2 | g = [−b : a : 0]; which is a single point in P2 on the line at infinity. n 1.1.4 Definition. If f 2 |[x1; : : : ; xn+1], then P = [a1 : ··· : an+1] 2 P is ∗ a zero of f if f(λa1; : : : ; λan+1) = 0 for every λ 2 | , in which case we write f(P ) = 0. For any S ⊆ |[x1; : : : ; xn+1], let n Vp(S) = fP 2 P j f(P ) = 0 for all f 2 Sg n n be the zero set of S in P . Moreover, if Y ⊆ P is such that Y = Vp(S) for some S ⊆ |[x1; : : : ; xn+1], then we say that Y is a projective algebraic set. Similarly, given Y ⊆ Pn, let Ip(Y ) = ff 2 |[x1; : : : ; xn+1] j f(P ) = 0 for all P 2 Y g be the ideal of Y . Remark. To avoid confusion, from now on we will use the notation Ia and n n Va for the ideal of a set of points in A and the zero set in A of a set of polynomials, respectively. 1.1.5 Lemma. Let f 2 |[x1; : : : ; xn+1] be such that f = fm + ··· + fd, where n each fi is an i-form. Then, for any P 2 P , f(P ) = 0 if and only if fi(P ) = 0 for i = m; : : : ; d. n Proof: Suppose f(P ) = 0 for P = [a1 : ··· : an+1] 2 P . Then m d q(λ) = λ fm(a1; : : : ; an+1) + ··· + λ fd(a1; : : : ; an+1) = 0 ∗ for all λ 2 | , and q is a polynomial in λ with coefficients bi = fi(a1; : : : ; an+1). Since | is infinite and q(λ) = 0, we have that bi = 0 for each i. Therefore, fi(λa1; : : : ; λan+1) = 0, and fi(P ) = 0 for each i. The converse is clear. Thus, if f = fm + ··· + fd 2 |[x1; : : : ; xn+1]; where each fi is an i-form, then Vp(f) = Vp(fm; : : : ; fd) and if f 2 Ip(Y ) for n some Y ⊆ P , then fi 2 Ip(Y ) for each i. 1.1.6 Proposition. (i) Every algebraic set in Pn is the zero set of a finite number of forms. n (ii) If Y ⊆ P , then Ip(Y ) is generated by homogeneous polynomials. Proof: The result is clear from the preceding discussion. 3 This motivates the following definition. 1.1.7 Definition. An ideal I ⊆ |[x1; : : : ; xn] is said to be homogeneous if whenever f 2 I then each homogeneous component of f is in I. n By the above, we see that Ip(Y ) is homogeneous for any Y ⊆ P . Moreover, one proves as in the affine case that Ip(Y ) is radical. We then have a corre- spondence between projective algebraic sets in Pn and homogeneous radical ideals in |[x1; : : : ; xn+1]. We will see that this is almost a one-to-one corre- spondence, but in order to make it one-to-one we have to exclude ? and the ideal of f(0;:::; 0)g. Let us begin by stating some properties of homogeneous ideals. 1.1.8 Proposition. Let I and J be ideals in |[x1; : : : ; xn+1]. Then: (i) I is homogeneous if and only if I can be generated by homogeneous polynomials; p (ii) if I and J are homogeneous, then I + J, IJ, I \ J, and I are homoge- neous; (iii) I is a homogeneous prime ideal if and only if whenever f; g 2 |[x1; : : : ; xn] are forms such that fg 2 I, then f 2 I or g 2 I. Proof: (i) Exercise (ii) Exercise. (iii) The forward direction is clear. Let f; g 2 |[x1; : : : ; xn+1] be such that Pd Pd0 fg 2 I. Let f = i=m fi and g = i=m0 gj, where each fi is an i-form and each gj is a j-form. Then d+d0 0 1 X X fg = fmgm0 + @ figiA : k>m+m0 i+j=k Since I is homogeneous, fmgm0 2 I. Suppose, for now, that fm 2= I, so that gm0 2 I. Then f(g−gm0 ) = (fg−fgm0 ) 2 I. Since the homogeneous 0 component of f(g − gm0 ) of degree m + m + 1 is fmgm0+1 and I is homogeneous, we have fmgm0+1 2 I. And since fm 2= I, this means that gm0+1 2 I so that f(g − gm0 − gm0+1) = fg − fgm0 − fgm0+1 2 I: Continuing this way, we see that if fm 2= I, then gi 2 I for all i = 0 0 m ; : : : ; d , implying that g 2 I. If both fm and gm0 are in I, proceed as above with (f − fm)(g − gm0 ). 1.1.9 Examples. (i) I = hx2i and J = hx2; yi are homogeneous in |[x; y]. 4 (ii) I = hx2 + xi is not homogeneous since x2 = I. 1.1.10 Definition. Let θ : An+1 n f0g ! Pn be the standard projection, so that θ(x1; : : : ; xn+1) = [x1 : ··· : xn+1]: If Y ⊆ Pn, the affine cone over Y is −1 n+1 C(Y ) = θ (Y ) [ f(0;:::; 0)g ⊆ A : We note the following properties of the affine cone. 1.1.11 Proposition. (i) If P 2 Pn, then C(fP g) is the line in An+1 through the origin determined by P . (ii) C(?) = f(0;:::; 0)g.
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