# Deligne–Mumford stack

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

• (i) the diagonal ${\displaystyle F\to F\times _{S}F}$ is representable (the base change to a scheme is a scheme), quasi-compact and separated.
• (ii) There is a scheme U and étale surjective map UF (called the 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 Artin stack. An algebraic space is Deligne–Mumford.

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

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