# How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on.

(1) Special case: Consider the natural ('change of variables') action of $G = GL(\mathbb{C}^3)$ on $\text{Sym}^3({\mathbb{C}^3}^*)$, the space of homogeneous cubic polynomials on $\mathbb{C}^3$. (Here, ${\mathbb{C}^3}^*$ denotes the dual space $\text{Hom}(\mathbb{C}^3,\mathbb{C})$.) With respect to the standard bases on $\mathbb{C}^3$ and ${\mathbb{C}^3}^*$, let $F = x_1^3 + x_2^3 + x_3^3 \in \text{Sym}({\mathbb{C}^3}^*)$.

Let $\phi : PGL(\mathbb{C}^3) \rightarrow \mathbb{P}(\text{Sym}^3({\mathbb{C}^3}^*))$ be the morphism given by $[g] \mapsto [g \cdot F]$. By dimension considerations, we see that the image of $\phi$ is a hypersurface in $\mathbb{P}(\text{Sym}^3({\mathbb{C}^3}^*)) \cong \mathbb{P}^9$.

Question (1): What is a concrete description of the image of $\phi$?

(2) General case: Consider an irreducible representation $W$ of $GL(V)$. Let $w \in W$ be a non-zero vector and define $\phi_w : PGL(V) \rightarrow \mathbb{P}(W)$ as above, $[g] \mapsto [g \cdot w]$.

Question (2): What are good references for learning about the geometry of the image of $\phi_w$?

In particular, I would ideally like references that deal with the case of a general $w$, not just for a (highest) weight vector. (The element $F$ in (1) is not a monomial, so not a weight vector.)

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Since $PGL$ acts transitively on the orbit of your $F$, a concrete description of the latter is as the quotient of $PGL$ by the stabilizer of $F$, which one can surely describe explicitlyâ€”it is in fact discrete. – Mariano Suárez-Alvarez May 11 '12 at 17:59