nLab projective stack

Definition

A projective space is a quotient

$$\mathbb{P}(X) := (X-\{0\})/\mathbb{G}_m$$

by the action of a multiplicative group object (for a given line object).

If instead of forming the quotient one forms the weak quotient/action groupoid, one speaks of the projective stack

$$\hat \mathbb{P}(X) := (X-\{0\})//\mathbb{G}_m \,.$$

References

The definition appears for instance as def 2.1.1 in

• Dan Abramovich and Brendan Hasset, Stable varieties with a twist (ps)