[Math] Automorphisms of a weighted projective space

Question

What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},…,a_{n})$ ?
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any automorphism has to fix the two singular points. Consider a smooth point $p\in\mathbb{P}(2,3,4)$. What is the subgroup of the automorphisms of $\mathbb{P}(2,3,4)$ fixing $p$ ?

Answer 1

The automorphism group is the quotient of the automorphism group of the corresponding graded algebra by 1-dimensional torus acting by rescaling. In the particular case of $P(2,3,4)$ the graded algebra is $A = k[x_2,x_3,x_4]$ with $\deg x_i = i$. Note that any automorphism should take $x_2 \to a x_2$, $x_3 \to b x_3$ (since those are only elements of $A$ of degree 2 and 3) and $x_4 \mapsto c x_4 + d x_2^2$. So, the group can be written as $((k^*)^3 \ltimes k) / k^*$, where $\ltimes$ stands for the semidirect product.