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 $Spec R$?

  • 1
    $\begingroup$ 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.) $\endgroup$ – David Zureick-Brown Jul 16 '13 at 17:22
  • 1
    $\begingroup$ Borisov-Chen-Smith, Abramovich-Hassett, Fantechi-Mann-Nironi, etc. $\endgroup$ – Jason Starr Jul 16 '13 at 17:29
  • $\begingroup$ @David:Thanks. Not taking the origin out was a typo. I just corrected it. $\endgroup$ – Lennart Meier Jul 17 '13 at 7:47

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.)


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

Perhaps Example $7.27$ of


could be interesting.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.