In algebraic geometry, a Deligne–Mumford stack is a stack F such that
- the diagonal morphism is representable, quasi-compact and separated.
- There is a scheme U and étale surjective map (called an atlas).
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 , where B is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.
Examples edit
Affine Stacks edit
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 on given by
Weighted Projective Line edit
Non-affine examples come up when taking the stack quotient for weighted projective space/varieties. For example, the space is constructed by the stack quotient where the -action is given by
Stacky curve edit
Non-Example edit
One simple non-example of a Deligne–Mumford stack is since this has an infinite stabilizer. Stacks of this form are examples of Artin stacks.