**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 $p$ ?** 



Automorphisms of a weighted projective space
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.


