derived smooth geometry
A Deligne-Mumford stack is the analogue in algebraic geometry of what in differential geometry is an orbifold: a stack quotient of a scheme over the étale site all whose automorphism groups are finite groups.
These are what originally were called algebraic stacks. The latter term nowadays often refers to the more general notion of Artin stack, where the automorphism groups are allowed to be more generally algebraic groups. This case is the algebraic version of the general notion of geometric stack.
Given a scheme . A -stack (i.e under the Grothendieck construction a category fibered in groupoids over satisfying descent) is Deligne-Mumford when it has a representable, separable and quasi-compact diagonal and a covering which is surjective, representable and etale, by an algebraic space .
DM-stacks are introduced in
Characterization of higher Deligne-Mumford stacks (see generalized scheme) are in