algebraic varieties
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| algebraic varieties | |
|---|---|
| Name | Algebraic varieties |
| Field | Algebraic geometry |
| Introduced | 19th century |
| Notable | Bernhard Riemann; Alexander Grothendieck; André Weil; Oscar Zariski; David Mumford |
algebraic varieties are geometric objects defined as the solution sets of systems of polynomial equations over a field. They form the central objects of study in algebraic geometry and connect to complex analysis, number theory, and topology through influential figures such as Bernhard Riemann, Alexander Grothendieck, André Weil, Oscar Zariski, and David Mumford. Historically developed through work on curves and elimination theory, varieties evolved into the scheme-theoretic language that underlies modern research influenced by institutions like the Institute for Advanced Study and the École Normale Supérieure.
Definition and basic examples
A basic definition presents a variety as the common zero locus of polynomials in affine or projective space over a field introduced by Carl Friedrich Gauss and generalized by Niels Henrik Abel and Évariste Galois. Classical examples include affine lines and conics studied by Johann Carl Friedrich Gauss, plane cubics connected to Srinivasa Ramanujan-related modularity phenomena, and nodal curves analyzed by Riemann in the context of his work on complex curves. Key simple instances are the affine line, plane curves like elliptic curves used by Andrew Wiles in the proof of the Taniyama–Shimura–Weil conjecture, and hypersurfaces such as Fermat varieties related to results of Pierre de Fermat and later developments by Gerd Faltings. The notion of a variety traditionally requires irreducibility and reduced structure, concepts clarified by the work of Oscar Zariski and formalized further by Grothendieck.
Affine and projective varieties
Affine varieties sit inside affine n-space, a setting implicit in the computations of David Hilbert’s basis theorems and the algorithms of Emmy Noether and Issai Schur. Projective varieties are embedded in projective space, enabling compactification techniques used by André Weil in formulating his axioms and by Alexander Grothendieck in constructing moduli spaces. The passage between affine and projective pictures is essential in the study of compactification phenomena appearing in the work of Shinichiro Mochizuki and in classical approaches to enumerative geometry pursued by Hermann Schubert and later by researchers at institutions such as the Institut des Hautes Études Scientifiques. Projective techniques are indispensable in results like the Riemann–Roch theorem for curves and its extensions by Jean-Pierre Serre and Armand Borel.
Morphisms, rational maps, and birational geometry
Morphisms between varieties capture algebraic maps studied by Alexander Grothendieck and formalized in scheme theory by Grothendieck and Jean-Pierre Serre. Rational maps, defined where polynomial expressions are not globally regular, are central in birational geometry developed through the work of Federigo Enriques, Francesco Severi, and modernized by Shigefumi Mori and collaborators. Birational equivalence classes underpin classification efforts exemplified by the Minimal Model Program advanced by figures associated with institutions like Princeton University and Harvard University. Techniques in resolving indeterminacy and studying exceptional loci appear in contributions by Heisuke Hironaka on resolution of singularities and in the empirical investigations led by researchers at the Mathematical Sciences Research Institute.
Dimension, singularities, and local properties
The dimension theory of varieties, refined by David Mumford and articulated in the language of Hilbert functions originating from David Hilbert, measures growth rates of coordinate rings and local parameter counts studied by Oscar Zariski. Singularities—points where local regularity fails—are central to work by John Milnor on singularity links and by Heisuke Hironaka on resolution techniques applied over fields of characteristic zero. Local rings and analytic germs connect algebraic varieties to complex analytic spaces considered by Kiyoshi Oka and global topological invariants examined in research related to the Atiyah–Singer index theorem by Michael Atiyah and Isadore Singer. Characteristic-p phenomena, including wild ramification and inseparable morphisms, were studied by Serre and later by researchers in arithmetic geometry at places like the University of Chicago.
Sheaves, schemes, and modern formulation
The reformulation of varieties within scheme theory by Alexander Grothendieck and Jean-Pierre Serre introduced sheaves, representable functors, and the functor-of-points perspective used in moduli problems addressed by David Mumford and others. Coherent sheaves and derived categories became essential in the work of Maxim Kontsevich and in homological approaches developed at research centers such as the Institut Henri Poincaré. The language of schemes unified arithmetic and geometric aspects pursued by Andre Weil in his conjectures and enabled proof strategies culminating in achievements by Pierre Deligne on the Weil conjectures. Stacks and algebraic spaces, advanced by researchers affiliated with institutions like Columbia University and University of California, Berkeley, extend the classical variety notion to parametrized families and quotient constructions.
Classification and examples (curves, surfaces, higher dimensions)
Classification begins with curves: genus stratification and the moduli space of curves studied by Riemann, Bernhard Riemann, and formalized by Mumford and researchers at the Institut des Hautes Études Scientifiques. Surfaces were systematically classified in the Enriques–Kodaira framework advanced by Federigo Enriques, Kunihiko Kodaira, and expanded in birational minimal model theory by Shigefumi Mori. Higher-dimensional classification remains active, with the Minimal Model Program and flips central to contemporary research influenced by seminars at Princeton University and collaborations involving mathematicians like James McKernan. Explicit families such as K3 surfaces connected to work by André Weil and Calabi–Yau varieties appearing in string-theoretic contexts studied at institutions like CERN provide rich examples. Arithmetic examples include rational points on varieties explored by Yuri Manin and diophantine geometry advances culminating in results by Gerd Faltings.