Deligne–Mumford stack
In algebraic geometry, a Deligne–Mumford stack is a stack F such that
- (i) the diagonal morphism is representable, quasi-compact and separated.
- (ii) There is a scheme U and étale surjective map U →F (called an atlas).
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 algebraic stack (also called an Artin stack). An algebraic space is Deligne–Mumford.
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.
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, MR 0262240, doi:10.1007/BF02684599
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