# 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.