Variety (mathematics)

Variety (mathematics)
Name	Variety (mathematics)
Field	Mathematics
Notable figures	Évariste Galois, David Hilbert, Alexander Grothendieck, Oscar Zariski, Serge Lang

Variety (mathematics) is a central object in modern Algebraic Geometry and related fields. It generalizes solution sets of polynomial equations studied by René Descartes, connects to concepts developed by Niels Henrik Abel, Bernhard Riemann, and bridges to work of Henri Poincaré, André Weil, and Jean-Pierre Serre. Varieties appear across theories pioneered by Felix Klein, David Mumford, and Alexander Grothendieck and are foundational in results like the Riemann–Roch theorem, the Weil conjectures, and the Modularity theorem.

Definition and basic examples

A variety is commonly defined as a geometric object locally modeled on the common zero loci of polynomial functions over a field such as Carl Friedrich Gauss's rational numbers or Isaac Newton's real numbers; classical examples include the affine line, projective space, conics, and elliptic curves studied by Niels Henrik Abel and Sophie Germain. Over Évariste Galois's fields, affine varieties and projective varieties provide the basic models for later constructions in the work of Oscar Zariski and David Hilbert. Concrete instances include plane curves like the circle studied by Euclid and the cubic curves central to Andrew Wiles's proof of the Taniyama–Shimura–Weil conjecture. Basic examples also connect to moduli spaces introduced by Alexander Grothendieck and expanded by David Mumford.

Algebraic varieties

Algebraic varieties are schemes of finite type over a field in the sense formalized by Alexander Grothendieck in the context of Grothendieck topology and Éléments de géométrie algébrique; this formalism refines classical conceptions by Jean-Pierre Serre and Oscar Zariski. Projective varieties are subvarieties of Projective space intimately linked to the Hilbert polynomial and constructions used by David Hilbert. Curves, surfaces, and higher-dimensional analogues relate to classification programs initiated by Federigo Enriques and advanced by Kunihiko Kodaira and Shigeru Iitaka. Important classes such as Fano varieties, Calabi–Yau varieties, and K3 surfaces appear in interactions with Edward Witten, Philip Griffiths, and Shing-Tung Yau.

Topological and differentiable varieties

In contrast to algebraic varieties, differentiable manifolds studied by Henri Poincaré and John Milnor share the name "variety" in some languages; the interplay between the De Rham theorem and Hodge theory developed by W.V.D. Hodge and Pierre Deligne links topological invariants of complex varieties to differential forms. Real algebraic varieties connect to results of Mikhail Gromov and Oleg Viro on topology of real loci, while the work of René Thom and Vladimir Arnold addresses singularities within differentiable settings. Tools from Morse theory and the study of characteristic classes by Raoul Bott and Shiing-Shen Chern apply to smooth varieties and complex manifolds encountered in Differential Geometry.

Morphisms, maps, and constructions

Morphisms of varieties generalize polynomial maps and are treated categorically in Grothendieck's framework; the language of functors and natural transformations used by Saunders Mac Lane and Samuel Eilenberg provides a structural backdrop. Important maps include birational maps central to the birational geometry program of Federigo Enriques and Shigefumi Mori, morphisms of moduli spaces as in work of David Mumford and Pierre Deligne, and morphisms used in descent theory by Alexander Grothendieck. Constructions such as blowing up, normalization, and fiber products were systematized in Grothendieck's theory and used by Oscar Zariski and Heisuke Hironaka in resolution of singularities contexts.

Dimension, singularities, and local properties

The notion of dimension for varieties, formalized using transcendence degree in classical algebraic geometry of Emmy Noether and David Hilbert, was extended via Krull dimension and scheme-theoretic notions by Alexander Grothendieck. Singularities are studied through local rings and resolutions; the resolution of singularities for characteristic zero by Heisuke Hironaka and developments in minimal model theory by Shigefumi Mori and Vyacheslav Shokurov are milestones. Local invariants such as multiplicity, tangent cones, and Milnor numbers link to work of John Milnor and Bernard Teissier, while duality theories by Alexander Grothendieck and Jean-Pierre Serre govern local cohomological properties.

Classification and major classes of varieties

Classification efforts include the birational classification of algebraic surfaces by Federigo Enriques and Kunihiko Kodaira and the higher-dimensional minimal model program led by Shigefumi Mori and Caucher Birkar. Major classes encompass curves classified by genus as in Bernhard Riemann's work, surfaces including K3 surfaces studied by Kunihiko Kodaira and Igor Shafarevich, Fano varieties central to Yuri Manin's conjectures, and Calabi–Yau varieties prominent in physical applications via Edward Witten and Cumrun Vafa. Moduli spaces such as the moduli of curves constructed by David Mumford and compactifications developed by Pierre Deligne and Gérard Laumon organize families of varieties and underpin modern arithmetic geometry pursued by Andrew Wiles and Richard Taylor.

Category:Algebraic geometry