Hartshorne 一章, 問題2.10

The Cone Over a Projective Variety. Let Y \subseteq \mathbb{P}^n be a nonempty algebraic set, and let \theta\colon \mathbb{A}^{n+1} - \{(0,\ldots,0)\} \to \mathbb{P}^n be the map which sends the point with affine coordinates (a_0,\ldots,a_n) to the point with homogeneous coordinates (a_0,\ldots,a_n). We define the affine cone over Y to be

C(Y) = \theta^{-1} \cup \{(0,\ldots,0)\}.

  1. Show that C(Y) is an algebraic set in \mathbb{A}^{n+1}, whose ideal is equal to I(Y), considered as an ordinary ideal in k[x_0,\ldots,x_n].
  2. C(Y) is irreducible if and only if Y is.
  3. \dim C(Y) = \dim Y + 1.

Sometimes we consider the projective closure \overline{C(Y)} of C(Y) in \mathbb{P}^{n+1}. This is called the projective cone over Y.

Proof of a. I(C(Y)) = I(\theta^{-1}(Y) \cup \{(0,\ldots,0)\}) = I(Y) since \theta^{-1}(Y) \cup \{(0,\ldots,0)\} consists of all scalar multiples of points in Y, i.e., all points in \mathbb{A}^{n+1} that are zeros of the generators of I(Y).

Proof of b. C(Y) is irreducible if and only I(C(Y)) is prime if and only if I(Y) is prime (by a) if and only if Y is irreducible.

Proof of c. \dim C(Y) = \dim k[x_0,\ldots,x_n]/I(Y) = \dim S(Y) = \dim Y + 1 by Problem 2.6.

Hartshorne 一章, 問題2.9

Projective Closure of an Affine Variety. If Y \subseteq \mathbb{A}^n is an affine variety, we identify \mathbb{A}^n with an open set U_0 \subseteq \mathbb{P}^n by the homeomorphism \varphi_0. Then we can speak of \overline{Y}, the closure of Y in \mathbb{P}^n, which is called the projective closure of Y.

  1. Show that I(\overline{Y}) is the ideal generated by \beta(I(Y)), using the notation of the proof of (2.2).
  2. Let Y \subseteq \mathbb{A}^3 be the twisted cubic of (Ex. 1.2). Its projective closure \overline{Y} \subseteq \mathbb{P}^3 is called the twisted cubic curve in \mathbb{P}^3. Find generators for I(Y) and I(\overline{Y}), and use this example to show that if f_1,\ldots,f_r generate I(Y), then \beta(f_1),\ldots,\beta(f_r) do not necessarily generate I(\overline{Y}).

Proof of a. Suppose F \in I(\overline{Y}). Then f = \varphi_0(F) = F(1,x_1,\ldots,x_n) vanishes on Y, so f \in I(Y). Since \beta(f) = F, we have that I(\overline{Y}) \subseteq \langle \beta(I(Y))\rangle. Similarly, if f \in \langle \beta(I(Y)) \rangle, then it is the sum of polynomials of the form \beta(F), each one of which is in I(\overline{Y}) after homogenization, and so \langle \beta(I(Y))\rangle \subseteq I(\overline{Y}).

Proof of b. Recall I(Y) = \langle z-x^3,y-x^2 \rangle. Now \overline{Y} = \{(1,s/t,s^2/t^2,s^3/t^3)\} = \{(t^3,st^2,s^2t,s^3)\}, and so I(\overline{Y}) = \langle x_0x_2 - x_1^2,x_0x_3 - x_1x_2,x_1x_3 - x_2^2\rangle, but this is not equal to \langle \beta(z-x^3),\beta(y-x^2) \rangle.

Hartshorne 一章, 問題2.8

A projective variety Y \subseteq \mathbb{P}^n has dimension n-1 if and only if it is the zero set of a single irreducible homogeneous polynomial f of positive degree. Y is called a hypersurface in \mathbb{P}^n.

Proof. \dim Y = n-1 \iff \dim S(Y) = n by Problem 2.6. By Proposition 1.13, this is true if and only if the affine cone is defined by as the zero set of a nonconstant irreducible polynomial f\in S, which is true if and only if Y is defined by the homogenization F of f.

Hartshorne 一章, 問題2.6

If Y is a projective variety with homogeneous coordinate ring S(Y), show that \dim S(Y) = \dim Y + 1.

Proof. Let \varphi_i\colon U_i \to \mathbb{A}^n be the homeomorphism of Proposition 2.2, let Y_i be the affine variety \varphi_i(Y \cap U_i), and let A(Y_i) be its affine coordinate ring. Assume i=0 is such that \dim Y_i = \dim Y. Any F/x_0^n \in S(Y)_{x_0} of degree zero can be written as F(1,x_1/x_0,\ldots,x_n/x_0) which is exactly \alpha(f) from Proposition 2.2. On the other hand any polynomial f \in A(Y_0) can be homogenized with F = \beta(f) from Proposition 2.2. If F has degree d, we associate F/x_0^d with it. Thus A(Y_0) can be identified with (S(Y)_{x_0})_0, and so S(Y)_{x_0} \cong A(Y_0)[x_0,x_0^{-1}]. Now the transcendence degree of K(A(Y_0)[x_0,x_0^{-1}]) over k is \dim Y_0 + 1 by Proposition 1.7 and Theorem 1.8A, and so \dim S(Y) = \dim S(Y)_{x_0} = \dim Y_0 + 1 = \dim Y + 1 by Problem 1.10.

Hartshorne 一章, 問題2.5

  1. \mathbb{P}^n is a noetherian topological space.
  2. Every algebraic set in \mathbb{P}^n can be written uniquely as a finite union of irreducible algebraic sets, no one containing another. These are called its irreducible components.

Proof. a follows from Problem 2.3 since k[x_0,\ldots,x_n] is noetherian. b is just Proposition 1.5.

Hartshorne 一章, 問題2.4

  1. There is a 1-1 inclusion-reversing correspondence between algebraic sets in \mathbb{P}^n, and homogeneous radical ideals of S not equal to S_+, given by Y \mapsto I(Y) and \mathfrak{a} \mapsto Z(\mathfrak{a}).
  2. An algebraic set Y \subseteq \mathbb{P}^n is irreducible if and only if I(Y) is a prime ideal.
  3. Show that \mathbb{P}^n itself is irreducible.

Proof. a is just Problem 2.3. b follows from Corollary 1.4 by looking at the affine cone. c follows since \mathbb{P}^n = Z(0).