Deligne–Mumford stack

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

the diagonal morphism $F\to F\times F$ is representable, quasi-compact and separated.
There is a scheme U and étale surjective map $U\to F$ (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 $F(B)$ , where B is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.

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