Deligne–Mumford stack

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

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.

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