Deligne–Mumford stack

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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 UF (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.

An important 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.