1
$\begingroup$

Let A consist of the columns of the $2\times (d+1)$ matrix

$$A=\begin{pmatrix} d & d-1 & \cdots & 1&0\\ 0 & 1 & \cdots & d-1 &d \end{pmatrix}$$

Then consider the map

\begin{array}{lcl}\theta_A:\mathbb{(C^*)}^2\rightarrow\mathbb{P}^d\\ (s,t)\rightarrow[s^d:s^{d-1}t:\dots:st^{d-1}:t^d]\end{array}

We know that $C_d$ is Zariski Closure of image of the $\theta_A$ and $C_d$ called the rational normal curve of degree d.

Now I want to show that $I(C_d)=\langle x_ix_{j+1}-x_{i+1}x_j: 0\le i <j\le d-1\rangle$.

Consider $$[s^d:s^{d-1}t:\cdots:st^{d-1}:t^d]=[1:\frac{t}{s}:\frac{t^2}{s^2}:\cdots:\frac{t^{d-1}}{s^{d-1}}:\frac{t^d}{s^d}]=[1:u:u^2:\cdots:u^{d-1}:u^d]$$ Then we get $B=\begin{bmatrix}0&1&\cdots&d-1&d\end{bmatrix}$. And also it is clear that $B$ is in the row space of $A$. This gives the map \begin{array}{lcl}\theta_B:\mathbb{(C^*)}^2\rightarrow\mathbb{P}^d\\ (s,t)\rightarrow[1:t:\dots:t^{d-1}:t^d]\end{array} And also I know $C_d$ is Zariski Closure of image of the $\theta_B$. But I dont know how can I show that $I(C_d)=\langle x_ix_{j+1}-x_{i+1}x_j: 0\le i <j\le d-1\rangle$. I need a hint for this.

I try to use Proposition 2.1.4(Cox,Little,Schenk-Toric Varieties) but I cant see yet.

$\endgroup$

0

You must log in to answer this question.