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# Contents

## Idea

A Deligne-Mumford stack (after Deligne-Mumford 69) is the analogue in algebraic geometry of what in differential geometry is an orbifold: a quotient stack 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 stacks, where the automorphism groups (isotropy groups) are allowed to be more general algebraic groups. This case is the algebraic version of the general notion of geometric stack.

## Definition

Given a scheme $S$. A $S$-stack $X$ (i.e under the Grothendieck construction a category fibered in groupoids over $(Aff/S)_{et}$ satisfying descent) is Deligne-Mumford when it has a representable, separable and quasi-compact diagonal $\Delta: X\rightarrow X\times_S X$ and a covering $P:A\rightarrow X$ which is surjective, representable and etale, by an algebraic space $A$.

Deligne-Mumford stacks correspond to moduli problems in which the objects being parametrized have finite automorphism groups. See also algebraic stack.

### As a generalized scheme

From the perspective of derived algebraic geometrys a Deligne-Mumford stack is a special case of a generalized scheme (or $G$-scheme for $G$ a geometry (for structured (∞,1)-toposes)) as follows:

###### Definition (etale geometry)

StSp Def 2.6.12

For $k$ a commutative ring, let the etale geometry $G_{et}(k)$ be the geometry (for structured (∞,1)-toposes) defined as follows:

• the underlying (∞,1)-category is is ordinary category

$G_{et}(k) := (CRing_k^{fin})^{op}$

of finitely presented affine schemes over $k$;

• a morphsim $f : Spec(A) \to Spec(B)$ is admissible precisely if the corresponding morphism $f^* : B \to A$ of commutative $k$-algebras is an etale map;

• the Grothendieck topology on $G_{et}(k)$ is the restriction of the standard etale topology.

###### Theorem

SrSp Thm. 2.6.16

An (∞,1)-presheaf $F : CRing_k \to \infty Grpd$ is a Deligne-Mumford stack precisely if it is representable by a $G_{et}(k)$-generalized scheme $(X,O_X)$ such that $X$ is 1-localic.

## In moduli space theory

An important source of DM-stacks are moduli problems, resulting often in moduli stacks (or their derived versions).

The concept is due to

Review (and relation to orbifolds):

• Andrew Kresch, On the geometry of Deligne-Mumford stacks (doi:10.5167/uzh-21342, pdf), in: D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) Algebraic Geometry: Seattle 2005, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 (pspum-80-1)

Characterization of higher Deligne-Mumford stacks (see generalized scheme) are in