(As suggested, this is a repost from [math.stackexchange.com](https://math.stackexchange.com/questions/4785/terminology-for-weighted-projective-spaces).)


For a sequence of positive integers $a\_1, \ldots, a\_n$ and a base ring $R$ there is a graded ring $R[x\_1,\ldots, x\_n]$ where $x\_i$ is in degree $a\_i$. We can then apply Proj and get a scheme, and this is usually called a weighted projective space; if all of the $a\_i$ are 1, then the resulting scheme really is projective space.


However, the way that this arises is as the quotient of $\mathbb{A}^n \setminus 0$ by an action of the multiplicative group, given by $(x\_1,\ldots,x\_n) \simeq (\lambda^{a\_1} x\_1, \ldots, \lambda^{a\_n} x\_n)$ for all $\lambda$. This is a "coarse" group quotient.


There is an alternative version where one instead takes the associated quotient stack/orbifold, and this has a number of nice properties (including possession of a line bundle $\mathcal{O}(1)$); this is true more generally of a graded ring.


It seems that most people associate "weighted projective space" with the scheme-theoretic notion. What is the appropriate terminology for the stack-theoretic version of this construction?



Terminology for weighted projective spaces
A google search for "[weighted projective stack](http://www.google.com/search?q=%22weighted+projective+stack%22)" currently yields 275 documents, many of which are quite legitimate.


