For $d\geq2$, the map
$\phi: \mathbb{C}^2 \rightarrow \mathbb{C}^{d+1}$
is defined by $\phi(s,t)=(s^d,s^{d-1}t,\dots,st^{d-1},t^d)$. Let $J_d=\langle x_ix_{j+1}-x_{i+1}x_{j}:0\le i<j\le d-1\rangle\subseteq \mathbb{C}[x_0,\dots,x_d]$. Then I want to show that the ideal of $\phi(\mathbb{C}^2)$ is $J_d$.
I have proved that $\phi(\mathbb{C}^2)=V(J_d)$. And it is well-known that $I(V(J_d))=\sqrt {J_d}$ by Hilbert's Nullstellensatz. Additionally if $J_d$ is prime or radical ideal then $\sqrt {J_d}=J_d$. I want to show that $J_d$ is a prime ideal so we have to say that $J_d$ is a toric ideal or $V(J_d)$ is an irreducible variety. Thus also we have to show that $V(J_d)$ is an affine toric variety, i.e. $V(J_d)$ has a torus. $\phi(\mathbb{C}^2)$ is a toric variety with torus $$\phi(\mathbb{C^*}^2)=\widehat{C_d}\cap (\mathbb{C^*})^{d+1}\simeq \mathbb{C^*}^2$$(see David Cox-Toric varieties pg.13).
How can I show that $\phi((\mathbb{C^*})^2) $ is the torus of $\phi(\mathbb{C}^2)$? I need some hint for this.