Deligne–Mumford stack

In algebraic geometry, a Deligne–Mumford stack is a stack F such that

Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks.

If the "étale" is weakened to "smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford.

A key fact about a Deligne–Mumford stack F is that any X in $$F(B)$$, where B is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.

Affine Stacks
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 $$C_n = \langle a \mid a^n =1 \rangle$$ on $$\mathbb{C}^2$$ given by $$a\cdot\colon(x,y) \mapsto (\zeta_n x, \zeta_n y).$$ Then the stack quotient $$[\mathbb{C}^2/C_n]$$ 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 $$(\text{Sch}/\mathbb{C})_{fppf}$$ then given a scheme $$S \to \mathbb{C}$$ the over category is given by $$ \text{Spec}(\mathbb{C}[s]/(s^n-1))\times \text{Spec}(\mathbb{C}[x,y])(S) \rightrightarrows \text{Spec}(\mathbb{C}[x,y])(S). $$ Note that we could be slightly more general if we consider the group action on $$\mathbb{A}^2 \in \text{Sch}/\text{Spec}(\mathbb{Z}[\zeta_n])$$.

Weighted Projective Line
Non-affine examples come up when taking the stack quotient for weighted projective space/varieties. For example, the space $$\mathbb{P}(2,3)$$ is constructed by the stack quotient $$[\mathbb{C}^2-\{ 0\}/\mathbb{C}^*]$$ where the $$\mathbb{C}^*$$-action is given by $$\lambda \cdot (x,y) = (\lambda^2x,\lambda^3y).$$ 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 $$(x,y) = (\lambda^2x,\lambda^3y)$$ if and only if $$x=0$$ or $$y=0$$ and $$\lambda = \zeta_3$$ or $$\lambda = \zeta_2$$, respectively, showing that the only stabilizers are finite, hence the stack is Deligne–Mumford.

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