Deligne–Mumford stack

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

(i) the diagonal morphism $F\to F\times F$ is representable, quasi-compact and separated.
(ii) There is a scheme U and étale surjective map U →F (called an atlas).

Deligne and 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). An algebraic space is Deligne–Mumford.

A key fact about a Deligne–Mumford stack F is that any X in $F(B)$ , B quasi-compact, has only finitely many automorphisms.

A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.

References

Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus", Publications Mathématiques de l'IHÉS, 36 (36): 75–109, MR 0262240, doi:10.1007/BF02684599

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