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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.

What is the appropriate terminology for the stack-theoretic version of this construction if "weighted projective space" is taken?

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Dear coarsemoduli, It seems to me quite likely that there is some standard terminology for this. If you don't get an answer here, you may want to ask on MO, where it more likely that your question will be seen by someone who knows what that standard terminology is. – Matt E Sep 16 '10 at 19:46
@Matt E: Thank you for your suggestion. I had considered posting it to MO, but it seemed that there was only one reason it might be more suitable there than here: more people on MO would know. – coarsemoduli Sep 17 '10 at 1:51

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