Hartshorne 一章, 問題2.12

The d-Uple Embedding. For given n,d > 0, let M_0,M_1,\ldots,M_N be all the monomials of degree d in the n+1 variables x_0,\ldots,x_n, where N = \binom{n+d}{n} - 1. We define a mapping \rho_d\colon \mathbb{P}^n \to \mathbb{P}^N by sending the point P = (a_0,\ldots,a_n) to the point \rho_d(P) = (M_0(a),\ldots,M_N(a)) obtained by substituting the a_i in the monomials M_j. This is called the d-uple embedding of \mathbb{P}^n in \mathbb{P}^N. For example, if n = 1, d = 2, then N = 2, and the image Y of the 2-uple embedding of \mathbb{P}^n in \mathbb{P}^N is a conic.

  1. Let \theta\colon k[y_0,\ldots,y_N] \to k[x_0,\ldots,x_n] be the homomorphism defined by sending y_i to M_i, and let \mathfrak{a} be the kernel of \theta. Then \mathfrak{a} is a homogeneous prime ideal, and so Z(\mathfrak{a}) is a projective variety in \mathbb{P}^N.
  2. Show that the image of \rho_d is exactly Z(\mathfrak{a}). (One inclusion is easy. The other will require some calculation).
  3. Now show that \rho_d is a homeomorphism of \mathbb{P}^n onto the projective variety Z(\mathfrak{a}).
  4. Show that the twisted cubic curve in \mathbb{P}^3 (Ex. 2.9) is equal to the 3-uple embedding of \mathbb{P}^1 in \mathbb{P}^3, for suitable choice of coordinates.

Proof of a. \mathfrak{a} is homogeneous since it is generated by homogeneous elements in k[y_0,\ldots,y_N], for each y_i is sent to an element of degree d. \mathfrak{a} is prime since \theta maps onto a subring of k[x_0,\ldots,x_n], which is an integral domain, so \mathfrak{a} is prime by the first isomorphism theorem.

Proof of b. f \in \mathfrak{a} \implies f(M_0,\ldots,M_N) = 0, so \mathrm{Im}\:\rho_d \subseteq Z(\mathfrak{a}). Conversely, Z(\mathfrak{a}) \subseteq \mathrm{Im}\:\rho_d \iff \mathfrak{a} \supseteq I(\mathrm{Im}\:\rho_d). If f \in I(\mathrm{Im}\:\rho_d), then it is identically zero for all Q \in \mathrm{Im}\:\rho_d, so f(M_0,\ldots,M_N) = 0, i.e. f \in \ker\theta.

Proof of c. By b it suffices to show that the map \rho_d is continuous with continuous inverse.

Proof of d. \mathbb{P}^1 \hookrightarrow \mathbb{P}^3 maps (t : s) \mapsto (t^3,t^2s,ts^2,s^3), which is the twisted cubic curve by Ex. 2.9b.

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