(As suggested, this is a repost from math.stackexchange.com.)

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?


A google search for "weighted projective stack" currently yields 275 documents, many of which are quite legitimate.

  • $\begingroup$ Yes, "weighted projective stack" has become standard. $\endgroup$ – Angelo Sep 25 '10 at 17:21

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