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Deligne-Mumford stack

Idea

A Deligne-Mumford stack is a stack (in the context of algebraic geometry) with the special property that it is covered, in a sense, by an ordinary algebraic space.

Definition

Given a scheme $S$. A $S$-stack $X$ (i.e under the Grothendieck construction a category fibered in groupoids over $(\mathrm{Aff}/S{)}_{\mathrm{et}}$ satisfying descent) is Deligne-Mumford when it has a representable, separable and quasi-compact diagonal $\Delta :X\to X{\times }_{S}X$ and a covering $P:A\to X$ 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 $G_{\mathrm{et}}(k)$ be the geometry (for structured (∞,1)-toposes) defined as follows:

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

  $$G_{\mathrm{et}}(k):=({\mathrm{CRing}}_k^{\mathrm{fin}}{)}^{\mathrm{op}}$$G_{et}(k) := (CRing_k^{fin})^{op}

  of finitely presented affine schemes over $k$;

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

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

Theorem

SrSp Thm. 2.6.16

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

DM stacks in moduli space theory

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

References

DM-stacks are introduced in

• P. Deligne, D. Mumford, “The irreducibility of the space of curves of given genus”. Publications Mathématiques de l’IHÉS (Paris) 36: 75–109 (1969) numdam