Find the action of $GL_3(K)$ on $\mathbb P_k^2 $, and compute its orbits and also the isotropy groups for all its orbits.
($K$ is an algebraically closed field)
I know that $GL_3$ acts on $P_k^2 $ via the standard action on $K^3$ but how can I compute the orbits and the isotropy groups?
I would really appreciate some help.
Observe that $\operatorname{GL}_3(K)$ acts transitively on $K^3\setminus\{0\}$ because for any nonzero $v\in K^3$, you can find an invertible linear map $\phi:K^3\to K^3$ that maps $(1,0,0)$ to $v$, say. Hence, the orbits in $K^3$ are $\{0\}$ and $K^3\setminus\{0\}$. In $\mathbb P^2_K$, there is no origin point, so the space itself is one big orbit.
To compute the stabilizer, you only have to compute the stabilizer of a single point. Indeed, let us compute the stabilizer $H$ of $[1:0:0]$. Then, $H$ consists of all invertible matrices that map the line $K\cdot(1,0,0)$ to itself. That would be the matrices $$\begin{pmatrix} x & a & b \\ 0 & h_{11} & h_{12} \\ 0 & h_{21} & h_{22} \end{pmatrix}$$ with $h\in\operatorname{GL}_2(k)$, $x\in K^\times$ and $a,b\in K$. This is a parabolic subgroup of $G$.