I am reading Lectures on Hilbert schemes by Nakajima, and this one argument really confuses me. For context I will lay out what is currently happening:
Let $V$ be a hermitian vector space, and $G$ a connected closed Lie subgroup of $U(V)$. Denote by $G^\mathbb{C}$ it's complexification, and fix a character: \begin{align*} \chi:G^\mathbb{C}\longrightarrow \mathbb{C}^* \end{align*} We can thus lift the $G^\mathbb{C}$ action to $V\times \mathbb{C}$ by defining: \begin{align*} g\cdot (v,z)=(g\cdot v,\chi(g)^{-1}\cdot z) \end{align*} We define the set of polynomials $A(V)^{G^\mathbb{C},\chi^n}$ as those which satisfy: \begin{align*} f(g\cdot v)=\chi(g)^nf(v) \end{align*} Then letting $\tilde{f}(v,z)=f(v)z^n$, we obtain a $G^\mathbb{C}$ invariant functions as: \begin{align*} \tilde{f}(g\cdot(v,z))=&\tilde{f}(g\cdot v,\chi(g)^{-1}(z))\\ =&f(g\cdot v)z^n\chi(g)^{-n}\\ =&f(v)z^n\\ =&\tilde{f}(v,z) \end{align*} We say that $v\in V$ is $\chi$-semi stable if there exists a polynomial $f\in A(V)^{G^\mathbb{C},\chi^n}$ such that $f(v)\neq 0$.
Nakajima then goes on to say that a point $v\in V$ is semi-stable if and only if the closure of the orbit $G^{\mathbb{\C}}\cdot (v,z)$ does not intersect with $V\times\{0\}$. He does not prove it just state it. So far I have tried this:
Suppose that the closure of the orbit of $(v,z)$ never intersects $V\times \{0\}$, then the orbit must never intersect $V\times \{0\}$ (I think this true? my point set topology has gotten a tad rusty), so it follows that for all $g\in G^{\C}$ $\tilde{f}(g\cdot (v,z))=f(v)z^n$, and that $z^n\neq 0$. How can I deduce that $f(v)\neq 0$?
For the other direction, suppose that $v\in V$ is semi-stable, then there exists an $f$ such that $f(v)\neq 0$. Consider the polynomial $\tilde{f}(v,z)$, with $z\neq 0$, then for all $g\in G^{\mathbb{\C}}$ we have that: \begin{align*} f(g\cdot(v,z))=f(v)z^n\neq 0 \end{align*} and I am not even sure how to relate this to the closure of the orbit to be honest.
For context, the whole point of this line of reasoning is to view the GIT quotient $V//_\chi G$ as a set theoretic quotient of the set of semi stable points.