Deligne–Mumford stack

Deligne–Mumford stack
Name	Deligne–Mumford stack
Field	Algebraic geometry
Introduced	1969
Introduced by	Pierre Deligne, David Mumford

Deligne–Mumford stack is a concept in algebraic geometry introduced by Pierre Deligne and David Mumford that refines the notion of moduli spaces, linking ideas from scheme theory, sheaf theory, and category theory. It provides a framework used by Grothendieck, Serre, Artin, and others to treat moduli problems with automorphisms, connecting to stacks, groupoids, and geometric invariant theory in the work of Mumford, Grothendieck, and Artin.

Definition and basic properties

A Deligne–Mumford stack is defined as an algebraic stack with unramified diagonal and admitting an étale surjective morphism from a scheme, following the formulations of Grothendieck, Artin, and Deligne in collaboration with Mumford and Serre; this definition relates to the formalism developed in Éléments de géométrie algébrique, SGA, and FGA under the influence of Weil and Zariski. The basic properties include the existence of étale atlases, finite inertia groups at geometric points described by Galois and Cartier-style finite group actions, and behavior under base change studied by Grothendieck, Artin, and Raynaud in the context of flatness and cohomology theories developed by Serre, Tate, and Faltings. Standard examples satisfy separatedness and properness criteria echoing results of Deligne, Mumford, Knudsen, and Harris and are compatible with intersection-theoretic frameworks by Fulton and MacPherson.

Examples and constructions

Typical examples include the moduli stack of smooth curves of genus g constructed by Deligne, Mumford, Harris, and Mumford’s compactification via stable curves studied by Knudsen and Hassett, as well as classifying stacks of finite groups related to Cartan and Weyl group actions appearing in work of Chevalley and Weyl. Quotient constructions arise from geometric invariant theory by Mumford, Kempf, and Ness and from orbifold quotients used by Thurston and McMullen in low-dimensional topology and dynamics; these constructions tie to toric geometry studied by Cox, Fulton, and Oda and to Shimura varieties investigated by Deligne, Milne, and Kottwitz. Other constructions are given by Hilbert schemes and Quot schemes of Grothendieck, Fogarty, and Simpson, and by stacky curves in the work of Behrend, Noohi, and Abramovich connecting to Kontsevich and Witten in enumerative geometry.

Moduli interpretations

Deligne–Mumford stacks often serve as fine moduli stacks parametrizing objects with finite automorphism groups, a perspective shaped by Grothendieck’s formulation of moduli, Tate’s study of abelian varieties, and the work of Mumford on theta functions; examples include moduli of curves (Deligne, Mumford, Harris), stable maps (Kontsevich, Manin), and principal bundles (Atiyah, Bott, Narasimhan, Seshadri). The moduli interpretation interfaces with Hodge theory of Griffiths and Schmid, with Torelli-type problems studied by Torelli, Torelli’s theorem contexts in work by Weil and Piatetski-Shapiro, and with arithmetic moduli considered by Faltings and Fontaine for applications to Diophantine geometry and the Langlands program pursued by Langlands, Drinfeld, and Lafforgue.

Geometric and cohomological properties

Geometric properties include notions of smoothness, properness, and separatedness paralleling those for schemes as in the work of Serre, Zariski, and Grothendieck, and they influence intersection theory developed by Fulton, Behrend, and Fantechi for virtual fundamental classes used by Kontsevich and Li. Cohomological aspects draw on étale cohomology by Grothendieck and Artin, on de Rham and Hodge cohomology studied by Deligne and Griffiths, and on derived categories influenced by Verdier, Bondal, Orlov, and Beilinson; these tools enable calculations of orbifold cohomology studied by Chen and Ruan and applications to mirror symmetry advanced by Kontsevich and Strominger–Yau–Zaslow.

Morphisms and substacks

Morphisms between Deligne–Mumford stacks generalize morphisms of schemes and are studied using 2-categorical techniques developed by Grothendieck, Giraud, and Artin; important classes include representable morphisms, étale morphisms, finite morphisms, and proper morphisms analyzed by Zariski, Nagata, and EGA authors. Substacks, including open and closed substacks, inertia stacks, and gerbes, are used in stratifications studied by Białynicki-Birula and Kirwan and in degeneration analyses by Deligne, Mumford, and Knudsen; these notions interact with resolution of singularities work by Hironaka and with semistable reduction results by Abramovich, Vistoli, and de Jong.

Relationship to Artin stacks and gerbes

Deligne–Mumford stacks sit as a full subcategory of Artin stacks (algebraic stacks) introduced by Artin and refined by Laumon and Moret-Bailly, with the defining distinction that inertia groups are finite and unramified, linking to gerbes banded by finite groups studied by Giraud, Breen, and Deligne. The passage from Deligne–Mumford to Artin settings is central in deformation theory initiated by Schlessinger and developed by Illusie, with gerbes and torsors playing roles in obstruction theories examined by SGA, Grothendieck, and Serre and in categorical formulations influenced by Mac Lane and Eilenberg.

Category:Algebraic geometry