Algebraic Variety
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| Algebraic Variety | |
|---|---|
| Name | Algebraic Variety |
| Field | Algebraic Geometry |
| Introduced | 19th century |
| Notable | Bernhard Riemann; Emmy Noether; André Weil; Alexander Grothendieck; David Mumford |
Algebraic Variety An algebraic variety is a fundamental geometric object defined by polynomial equations over a field, central to classical and modern Algebraic Geometry research. Varieties bridge concrete problems in Gauss-era algebra with abstract frameworks developed by Galois, Riemann, Hilbert, and Grothendieck and underpin advances in Weil’s conjectures, Deligne’s proofs, and modern work by Serre and Tate.
Definition and Basic Examples
A variety is locally modelled on zero loci of polynomial systems; classical examples include plane curves like the conic defined by a quadratic, elliptic curves studied by Abel and Jacobi, and higher-dimensional hypersurfaces such as the cubic and the K3. Early work by Riemann on Riemann surfaces inspired the interpretation of complex plane curves as compact one-dimensional varieties, while Weil and Zariski refined the algebraic foundations. Examples connecting arithmetic include the Mordell-type curves considered by Faltings and rational points on varieties studied by Manin and Lang.
Affine and Projective Varieties
Affine varieties arise as solution sets in Affine space cut out by polynomial ideals, with coordinate rings studied by Noether and Hilbert’s Nullstellensatz linking geometry to algebra. Projective varieties live in Projective space to compactify affine ones; classical projective curves include the plane cubic and the sphere viewed via stereographic projection. The passage between affine and projective settings is central in the work of Enriques, Kodaira, and Yau on classification, and underlies constructions in Birational geometry pursued by Mori and Voevodsky.
Morphisms, Rational Maps, and Functions
Morphisms between varieties are maps given locally by polynomials; they generalize classical parametrizations such as the rational parametrization of the Circle and the Rational maps used by Zariski in birational classification. Rational maps, regular functions, and field of rational functions are tools developed by Grothendieck and Serre to formalize maps defined only on open subsets, employed in proofs by Weil and Grothendieck for cohomological methods. Key figures like Mumford advanced geometric invariant theory relating morphisms to moduli problems studied by Artin and Laumon.
Dimension, Singularities, and Smoothness
Dimension theory for varieties, treated by Krull and Hilbert, measures degrees of freedom and links to transcendence degree in works by Artin. Singularities—points where local rings fail to be regular—were intensively studied by Zariski, Hironaka (resolution of singularities), and Mumford; Hironaka’s theorem over characteristic zero and ongoing problems in positive characteristic relate to research by Nagata and Mori. Smooth (non-singular) varieties connect to differential geometry via Chern and Kodaira, with links to classification efforts by Enriques and the Enriques–Kodaira classification.
Schemes and Modern Generalizations
Grothendieck’s introduction of schemes generalized varieties by allowing nilpotents and broader base rings, enabling deep results like the proof of the Weil conjectures by Deligne and providing the language for Étale cohomology developed by Grothendieck and Serre. Stacks, formal schemes, and derived schemes are further extensions used in moduli theory by Lieblich, Lurie, and Toën. The scheme framework unified arithmetic and geometric approaches pursued by Tate, Deligne, and Grothendieck.
Key Theorems and Properties
Central results include Hilbert’s Nullstellensatz connecting ideals to varieties, the Riemann–Roch theorem for curves and surfaces refined by Atiyah and Singer and Atiyah, Hironaka’s resolution of singularities, Zariski’s main theorem, and the Lefschetz hyperplane theorem with applications by Green and Griffiths. Theorems by Weil, Deligne, and Grothendieck on cohomology and L-functions link geometry to arithmetic studied by Frey and Wiles.
Applications and Examples in Geometry and Number Theory
Varieties model solutions to Diophantine equations central to Fermat’s Last Theorem proved by Wiles, while elliptic curves and modular curves connect to results by Faltings and Mazur on rational points and torsion. Calabi–Yau varieties are pivotal in string theory research by Witten and Candelas, and Shimura varieties link automorphic forms studied by Shimura and Yoshida to arithmetic geometry. Real and complex algebraic varieties appear in classification projects by Kodaira and moduli problems addressed by Mumford and Deligne.