Given the veronese mapping for n=1 \begin{align*} v_d:\mathbb{P}^1(\mathbb{C})&\longrightarrow\mathbb{P}^d(\mathbb{C})\\ \langle z_0,z_1\rangle&\longmapsto\langle z_0^d,z_0^{d-1}z_1,\cdots,z_1^d\rangle \end{align*} my goal is to show that
$$v_d(\mathbb{P}^1(\mathbb{C}))=Z(\langle x_ix_j-x_{j-1}x_{i+1}\rangle).$$
The inclusion $\subset$ is clear.
The other inclusion however I need a little help.
Attempt: we let $\langle x_0,\cdots ,x_d\rangle\in \mathbb{P}^d$, which satisfies $x_ix_j-x_{j-1}x_{i+1}=0$. Then let $i$ such that we always assume $x_i$ is non zero. Since $x_i^2=x_{i-1}x_{i+1}$, we may assume by induction that either $i=0$ or $i=d$.
For $i=0$: It follows that $x_j = x_{j-1}\frac{x_1}{x_0}$ , for $1\leq j \leq d$.
So we let $z=\frac{x_1}{x_0}$ and it follows that \begin{align*} \langle x_0,\cdots, x_d\rangle &= \langle x_0,zx_0,z^2x_0,\cdots,z^dx_0\rangle\\ &=v_d(\langle 1,z\rangle) \end{align*}
Could anyone tell me how to then proceed with the case $i=d$?
Thanks in advance!