moduli stacks

moduli stacks
Name	Moduli stacks
Field	Algebraic geometry
Introduced	1960s
Notable	Artin, Deligne, Mumford, Grothendieck

moduli stacks

Moduli stacks are central objects in algebraic geometry developed by Alexander Grothendieck, Michael Artin, Pierre Deligne, and David Mumford. They refine notions introduced in Federigo Enriques's and Oscar Zariski's era and connect to work by André Weil, Jean-Pierre Serre, and John Tate. Moduli stacks unify constructions appearing in the work of Igor Shafarevich, Alexander Grothendieck's school at the Institut des Hautes Études Scientifiques, and later developments by Maxim Kontsevich, Edward Witten, and Barry Mazur.

Introduction

Moduli stacks formalize parameter spaces pioneered by Bernard Riemann, David Hilbert, Felix Klein, and Henri Poincaré, extending ideas from Oscar Zariski and Federigo Enriques. The framework relies on representability criteria of Alexander Grothendieck and algebraization results due to Jean-Pierre Serre and Grothendieck's EGA. Foundational existence theorems were proved by Michael Artin and Pierre Deligne in correspondence with the moduli problems studied by David Mumford, Igor Dolgachev, and André Weil.

Definitions and Basic Properties

A moduli stack is defined using the language of stacks developed by Grothendieck and codified by Jean Giraud in the context of nonabelian cohomology. Deligne and Mumford introduced the notion of a \'Deligne–Mumford stack\' inspired by work on curves by Bernard Riemann and compactifications by Henri Poincaré; Artin formulated criteria for algebraic stacks building on representability conditions from Michael Artin and deformation theory of Alexander Grothendieck. The definition employs fibered categories over the category used by Grothendieck in EGA and gluing techniques akin to early work by Oscar Zariski and further elaborations by Alexander Grothendieck in SGA. Coarse moduli spaces relate to earlier notions studied by David Mumford via geometric invariant theory of David Hilbert and compactifications motivated by John Nash and later refined by Maxim Kontsevich.

Examples and Key Moduli Stacks

Fundamental examples include stacks parametrizing algebraic curves studied by Bernard Riemann, whose compactifications led to the Deligne–Mumford stack constructed by Pierre Deligne and David Mumford. The stack of vector bundles on curves draws on ideas from Atiyah Bott and links to gauge theory of Simon Donaldson and Edward Witten. Moduli of principal bundles relate to Claude Chevalley's and Armand Borel's structure theory for algebraic groups and to work by Michael Atiyah and Isadore Singer. Stacks of stable maps originate in enumerative geometry developed by Maxim Kontsevich and intersect the quantum field insights of Edward Witten and the mirror symmetry program initiated by Philip Candelas. Hilbert and Quot schemes of David Hilbert and Alexander Grothendieck appear as substacks linked to the constructions by Daniel Quillen and Pierre Deligne. The Picard stack connects to the classical theory of Jacobians explored by Friedrich Schottky and Carl Gustav Jacob Jacobi and to moduli of sheaves studied by Maruyama, Le Potier, and Simpson.

Construction Techniques

Techniques include Artin's representability criteria developed by Michael Artin using deformation theory influenced by Alexander Grothendieck and obstruction theories formalized by Michael Artin and Pierre Deligne. Geometric invariant theory by David Mumford provides quotient constructions used by David Hilbert and John Nash-inspired compactifications. Étale descent and fppf topology methods trace back to Alexander Grothendieck's work in SGA and to descent theory of Jean Giraud. Stacky approaches exploit groupoid presentations reminiscent of Lie groupoid techniques used in the analytic work of Charles Ehresmann and tied to orbifold concepts popularized by William Thurston. Derived enhancements leverage techniques from Pierre Deligne's cohomological theories and the derived algebraic geometry program advanced by Jacob Lurie and Bertrand Toën with contributions by Gabriele Vezzosi.

Geometric and Cohomological Properties

Geometric properties of stacks—smoothness, properness, and compactness—are analyzed using methods of Alexander Grothendieck's cohomology and duality and via compactification techniques by David Mumford and Pierre Deligne. Intersection theory on stacks extends the work of William Fulton and connects to virtual fundamental classes introduced by Kai Behrend and Barbara Fantechi. Cohomological field theories relate stacks to the enumerative frameworks developed by Maxim Kontsevich, Yuri Manin, and Edward Witten. The study of Picard and Brauer groups of stacks draws on contributions by Grothendieck, Jean-Pierre Serre, and Alexander Grothendieck's analyses in SGA while monodromy and local systems reflect influences from Bernard Riemann and Henri Poincaré as reformulated by Pierre Deligne.

Applications and Relationships to Other Moduli Spaces

Moduli stacks interface with moduli of varieties studied by Igor Shafarevich and birational geometry advanced by Shigefumi Mori and Yujiro Kawamata; they inform compactifications akin to those by David Mumford and Yum-Tong Siu. Connections to mathematical physics arise through work by Edward Witten, Maxim Kontsevich, and Anton Kapustin in gauge theory and string theory. Arithmetic applications engage research of Barry Mazur, Jean-Pierre Serre, and Pierre Deligne in Galois representations and Shimura varieties studied by Goro Shimura and Yutaka Taniyama. Interactions with categorical methods appear in the work of Alexander Beilinson, Joseph Bernstein, Maxim Kontsevich, and Jacob Lurie, linking stacks to derived categories explored by Amnon Neeman and Alexander Bondal.

Category:Algebraic geometry