Weighted projective spaces as stacks As stacks are the weighted projective line $\mathbb{P}$(1,n-1) and $\mathbb{P}$(k,n-k) isomorphic? Is there any reference for this?
 A: Let $X$ be a weighted projective (stacky) line. At least if we work over a field $K$, the Picard group of $X$ is $\mathbb{Z}$. Only powers of one of the two generators have global sections. Call this generator $\mathcal{O}(1)$ and its powers $\mathcal{O}(l)$. Thus, we can attach to $X$ the graded ring $R_*(X) =\bigoplus_{l\in\mathbb{Z}}H^0(X; \mathcal{O}(l))$. 
We have $R_*(\mathbb{P}(n,k)) = K[x,y]$ with $|x| = n$ and $|y| = k$. Thus, we can recover the degrees $n$ and $k$ from the graded ring $R_*(X)$. In particular, the weights $k$ and $n$ are uniquely determined.
A: This is not true, and you can verify that it is not so by checking the group of automorphisms of the $k$-points for $k$ a field. In the case of $\Bbb P(1,n-1)$, the inclusion of graded rings $k[x,y^{n-1}] \to k[x,y]$ (where $|y| = 1$ and $|x| = n-1$) gives a projection down to a coarse moduli space $\Bbb P^1$. The stabilizers of the points where $y \neq 0$ are all trivial (there is an open embedding $\Bbb A^1 \to \Bbb P(1,n-1)$ with this image), but the stabilizer of the point where $y=0$ is the cyclic group of $(n-1)$'st roots of unity.
By contrast, $\Bbb P(2,2)$ also has coarse moduli space $\Bbb P^1$ but the automorphism group of any point is $\{\pm 1\}$.
