Deligne–Mumford stack

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 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 ${\displaystyle F(B)}$, 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, doi:10.1007/BF02684599, MR 0262240