The standard action of $\operatorname{GL_2} K$ on $V$ induces an action of $\operatorname{PGL_2} K$ on $\mathbb{P}(\operatorname{Sym}^4 V)$. So far, I understood how all the orbits can be obtained via the $j$-function, but I'm struggling to see some algebraic properties of the more obvious orbits.
Specifically, define
$C = \{ [v^4] | v \in V\}$
$\Sigma = \{ [v^3\cdot w] | v, w \operatorname{independent} \in V\}$
$\Phi = \{ [v^2\cdot w^2] | v, w \operatorname{independent} \in V\}$
$\Psi = \{ [v^2\cdot w \cdot u] | v, w,u \operatorname{pairwise independent} \in V\}$
Now these are orbits because three pairwise independent vectors are projectively equivalent in $\mathbb{P}^1$, and C is a closed orbit because it can be described as a rational normal curve of degree 4.
Supposedly, this is the only closed orbit of all the above. How, for instance, can I see that $\Sigma$ is not closed in the Zariski topology? (So far, I have never computed why some set is NOT a variety)
I'd appreciate any help.
Edit: K is assumed to be algebraically closed.