In algebraic geometry, a Deligne–Mumford stack is a stack F such that
- (i) the diagonal 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).
A key fact about a Deligne–Mumford stack F is that any X in , B quasi-compact, has only finitely many automorphisms.
A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme.
- 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
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