Deligne–Mumford stack

Deligne–Mumford stacks are typically constructed by taking the stack quotient of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group ${\displaystyle C_{n}=\langle a\mid a^{n}=1\rangle }$ on ${\displaystyle \mathbb {C} ^{2}}$ given by

$${\displaystyle a\cdot \colon (x,y)\mapsto (\zeta _{n}x,\zeta _{n}y).}$$

Then the stack quotient ${\displaystyle [\mathbb {C} ^{2}/C_{n}]}$ is an affine smooth Deligne–Mumford stack with a non-trivial stabilizer at the origin. If we wish to think about this as a category fibered in groupoids over ${\displaystyle ({\text{Sch}}/\mathbb {C} )_{fppf}}$ then given a scheme ${\displaystyle S\to \mathbb {C} }$ the over category is given by

$${\displaystyle {\text{Spec}}(\mathbb {C} [s]/(s^{n}-1))\times {\text{Spec}}(\mathbb {C} [x,y])(S)\rightrightarrows {\text{Spec}}(\mathbb {C} [x,y])(S).}$$

Note that we could be slightly more general if we consider the group action on ${\displaystyle \mathbb {A} ^{2}\in {\text{Sch}}/{\text{Spec}}(\mathbb {Z} [\zeta _{n}])}$.

Non-affine examples come up when taking the stack quotient for weighted projective space/varieties. For example, the space ${\displaystyle \mathbb {P} (2,3)}$ is constructed by the stack quotient ${\displaystyle [\mathbb {C} ^{2}-\{0\}/\mathbb {C} ^{*}]}$ where the ${\displaystyle \mathbb {C} ^{*}}$-action is given by

$${\displaystyle \lambda \cdot (x,y)=(\lambda ^{2}x,\lambda ^{3}y).}$$

Notice that since this quotient is not from a finite group we have to look for points with stabilizers and their respective stabilizer groups. Then ${\displaystyle (x,y)=(\lambda ^{2}x,\lambda ^{3}y)}$ if and only if ${\displaystyle x=0}$ or ${\displaystyle y=0}$ and ${\displaystyle \lambda =\zeta _{2}}$ or ${\displaystyle \lambda =\zeta _{3}}$, respectively, showing that the only stabilizers are finite, hence the stack is Deligne–Mumford.

One simple non-example of a Deligne–Mumford stack is ${\displaystyle [pt/\mathbb {C} ^{*}]}$ since this has an infinite stabilizer. Stacks of this form are examples of Artin stacks.