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.

The group Aut(WPS) is studied in "Classes d'idéaux et Groupe de Picard des Fibrés projectifs tordus " §8.2.,8.3.

K-Theory 2 (1989),559-578 (A. AL-AMRANI)

A. Al-Amrani, May 16, 2013.

Just to complement Al-Amrani's answer (and to self-promote a little bit): the automorphism 2-group of the weighted weighted projective stack $\mathbb{P}_S(a_1,\ldots,a_n)$ over an arbitrary base scheme $S$ has been computed, and its structure elaborated, in https://doi.org/10.1016/j.jalgebra.2017.05.002

If I am not misunderstanding something, Al-Amrani's result can be interpreted as saying that the $\pi_0$ of the 2-group of automorphisms of the stack is isomorphic to the group of automorphisms of its coarse moduli space (namely, the corresponding weighted projective space).