Let $p=[0,\cdots,0,1]\in \mathbb{P}^n$, $X \subseteq \mathbb{P}^n-\{p\}$ a projective variety and let $\overline{X}$ be the projection of from $p$ to $V(X_n)\cong \mathbb{P}^{n-1}$. I know that $\overline{X}$ is a projective variety using resultant. Let $J=I(X)\cap \mathbb{C}[X_0,\cdots,X_{n-1}]$. I want to show that $V(J)=\overline{X}$ and so $I(\overline{X})=I(X)\cap \mathbb{C}[X_0,\cdots,X_{n-1}]$.
I already show that $\overline{X} \subseteq V(J)$. but, other side I do not show... is it possible??
