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For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on $Spec R[x_0,\dots, x_n] = \mathbb{A}^{n+1}_R$ induced by

$R[x_0,\dots, x_n] \to R[x_0,\dots, x_n] \otimes \mathbb{Z}[x,x^{-1}], x_i \mapsto x_i \otimes t^{a_i}$

The stack quotient $\mathcal{P}_R(a_0,\dots, a_n) = [\mathbb{A}^{n+1}_R-\{0\}/\mathbb{G}_m]$ is a called a weighted projective stack. For $a_0 = \cdots = a_n = 1$, one gets back the usual projective space.

This seems to be an important example of an algebraic stack, but I am unable to find a reference where basic facts about weighted projective stacks are proven. For example: Is there a reference where it is proven that $\mathcal{P}_R(a_0,\dots, a_n)$ is proper and smooth over $Spec R$?

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I don't know a reference, but want to point out that the stack quotient of a smooth scheme by a smooth group scheme is always smooth. Also, if you do not pop out the origin, then the stack quotient is not proper, since the origin has a Gm stabilizer. (The diagonal is then not proper, so the stack is not separated.) – David Zureick-Brown Jul 16 '13 at 17:22
Borisov-Chen-Smith, Abramovich-Hassett, Fantechi-Mann-Nironi, etc. – Jason Starr Jul 16 '13 at 17:29
@David:Thanks. Not taking the origin out was a typo. I just corrected it. – Lennart Meier Jul 17 '13 at 7:47
up vote 7 down vote accepted

I don't know a reference, but I think you can make the following argument.

The weighted projective stack $X$ is a $\mathbb{G}_m$-quotient of a scheme smooth over $R$, so it must be smooth over $R$ too: smoothness descends under flat morphisms, in the sense that if $Z \to W$ is faithfully flat and $Z$ is smooth, then so is $W$. (Here the map is a $\mathbb{G}_m$-torsor, so one doesn't need this stronger statement.)

For properness, you can use the valuative criterion (in the stacky version, as in Deligne-Mumford's paper on moduli of curves). Namely, assume that $R$ is a complete discrete valuation ring with residue field $K$. Then there are two things to show:

  • The restriction from $R$-points of $X$ to $K$-points of $X$ is fully faithful (as a functor of groupoids).
  • Given a $K$-point of $X$, after a ramified base extension to $K'$, it extends to an $R'$-point (where $R'$ is the integral closure in $K'$).

Over a local ring $R$, the groupoid of $X$-valued points is obtained by taking the set of tuples $\{x_0, \dots, x_n\}$ (at least one of which is invertible) and quotienting by the action of the group $R^*$ (with the weighted action). There is no further sheafification needed since the Picard group is trivial. It follows that given two $R$-valued points, an isomorphism between the associated $K$-valued points necessarily comes from an isomorphism of $R$-valued points: the $K$-scalar inducing the isomorphism must be a unit.

Conversely, given a $K$-valued point $\{x_0, \dots, x_n\}$, then it is isomorphic to an $R$-valued point if the valuation of each $x_i$ is divisible by $a_i$, and after a ramified base change you can assume this. (This part, which uses the "stacky" version of the valuative criterion, is not necessary to run the argument for ordinary projective space.)

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Weighted projective stacks could be characterized as complete toric orbifolds with cyclic Picard group.

Perhaps Example $7.27$ of

could be interesting.

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