Deligne–Mumford stack

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

• (i) the diagonal $F \to F \times_S F$ 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 U →F (called the atlas).

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 $F(B)$, B quasi-compact, has only finitely many automorphisms.

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