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<div>In algebraic geometry, a '''Deligne–Mumford stack''' is a [[stack (mathematics)|stack]] ''F'' such that<br />
*(i) the diagonal <math>F \to F \times_S F</math> is [[representable morphism of stacks|representable]] (the base change to a scheme is a scheme), quasi-compact and separated.<br />
*(ii) There is a scheme ''U'' and [[étale morphism|étale]] subjective map ''U'' →''F'' (called the atlas).<br />
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If the "étale" is weakened to "[[smooth morphism|smooth]]", then such a stack is called an [[Artin stack]]. An [[algebraic space]] is Deligne–Mumford.<br />
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An important fact about a Deligne–Mumford stack ''F'' is that any ''X'' in <math>F(B)</math>, ''B'' quasi-compact, has only finitely many automorphisms.<br />
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== References ==<br />
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Mumford | first2=David | author2-link=David Mumford | title=The irreducibility of the space of curves of given genus | url=http://www.numdam.org/item?id=PMIHES_1969__36__75_0 | id={{MathSciNet | id = 0262240}} | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | issue=36 | pages=75–109 | doi=10.1007/BF02684599 | volume=36}}<br />
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{{DEFAULTSORT:Deligne-Mumford stack}}<br />
[[Category:Algebraic geometry]]<br />
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{{geometry-stub}}</div>141.2.248.133