I am trying to solve: Consider the conic $C = Z(X_{0}X_{1} - X_{2}^{2})$ in the projective plane.
(a) Find the pointwise stabiliser of $C$ in $PGL(3,K)$
(b) Find the setwise stabiliser of $C$ in $PGL(3,K)$
My attempt:
Step1: I try to find the points in $C$ as follows:
We have two cases to consider:
Case1: $X_{0} = 0$. In this case, we must have $X_{2} = 0$, then we just have one point on the conic of the form $(0,a,b)$ which is $(0,1,0)$.
Case2: $X_{0} \neq 0$. In this case, we have all the points of the form $(1,a,a^{2})$ where a is any element of $K$, including $0$.
Step2:
Then, I take any 3 x 3 matrix, right now, I have no idea about the entries of the matrix. Since we are looking for the pointwise stabiliser of the conic, our matrix must map $[0,1,0]$ to itself. This means that, the second column of the matrix must be $[0,\lambda,0]$ for some nonzero $\lambda$ in $K$.
If we apply the same reasoning to $[1,0,0]$, then the first column of the matrix must be $[\beta,0,0]$.
Since our matrix, must be invertible, the last row cannot be $[0,0,0]$ thus it is of the form $[0,0,\alpha]$ for some nonzero $\alpha$ in $K$. Since, we are working with projectivities, we can normalize the last row to $[0,0,1]$. Now, our matrix looks like: \begin{bmatrix} \beta & 0 & *\\ 0 & \lambda & *\\ 0 & 0 & 1 \end{bmatrix} where * is any element of $K$, including $0$. I came here so far.
My main question is how to solve such questions in general: We are given an object in projective space, not necessarily zero locus of a linear form. How to find its pointwise and setwise stabiliser in projectivity group? Is there an algorithm to do this?
Thanks, in advance.