nLab Deligne-Mumford stack




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.


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

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 GG-scheme for GG a geometry (for structured (∞,1)-toposes)) as follows:

Definition (etale geometry)

StSp Def 2.6.12

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


SrSp Thm. 2.6.16

An (∞,1)-presheaf F:CRing kGrpdF : CRing_k \to \infty Grpd is a Deligne-Mumford stack precisely if it is representable by a G et(k)G_{et}(k)-generalized scheme (X,O X)(X,O_X) such that XX 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 (cf. also algebraic stack) is due to:

Review in the general context of algebraic stacks:

and with emphases on the 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 at generalized scheme):

Last revised on April 16, 2023 at 13:40:05. See the history of this page for a list of all contributions to it.