DeligneMumford

Deligne-Mumford stacks, named after Pierre Deligne and David Mumford, are a class of algebraic stacks that generalize moduli spaces to allow objects with finite automorphisms. They arose from the study of moduli problems, notably the moduli of algebraic curves, where naive spaces fail to be schemes or algebraic spaces due to nontrivial automorphisms.

A Deligne-Mumford stack X over a base scheme S is an algebraic stack with two key properties: first, the inertia (or stabilizer) groups of geometric points are finite, equivalently the diagonal morphism X -> X ×_S X is unramified; second, there exists an étale surjective morphism from a scheme U to X, called an atlas, so that X is covered locally by schemes in the étale topology. These conditions ensure that X behaves like a “moduli space up to finite automorphisms.”

Consequences and features include that X has finite automorphism groups at geometric points, and that many moduli problems can be formulated as Deligne-Mumford stacks. Deligne-Mumford stacks typically admit a coarse moduli space M, an algebraic space (often a scheme in nice cases) that corepresents isomorphism classes of objects. Proper DM stacks frequently yield proper coarse moduli spaces.

Examples include the classifying stack BG of a finite group G, the quotient stack [X/G] for a finite group G acting on a scheme X, and the moduli stacks of curves: M_g for smooth genus g curves (g ≥ 2), M_{g,n} for n-pointed curves, and its compactification \overline{M}_{g,n} of stable curves. DM stacks provide a robust framework for moduli problems with finite automorphisms and are a central object in modern algebraic geometry, sitting between schemes and more general Artin stacks.