Projective variety
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| Projective variety | |
|---|---|
| Name | Projective variety |
| Field | Algebraic geometry |
| Introduced | 19th century |
| Key figures | Felix Klein; Bernhard Riemann; Alexander Grothendieck; David Hilbert; Emmy Noether |
Projective variety A projective variety is a closed, algebraic subset of projective space defined by homogeneous polynomial equations. It is a central object in algebraic geometry and connects classical work by Bernhard Riemann, Felix Klein, and David Hilbert with modern schemes developed by Alexander Grothendieck and categorical methods related to Emmy Noether. Projective varieties serve as compact analogues of affine varieties and interact with topics such as divisors, cohomology, and moduli problems studied by researchers at institutions like the Institute for Advanced Study and universities such as Princeton University and University of Göttingen.
Definition and basic properties
A projective variety is defined inside projective space P^n over a field by the vanishing of a family of homogeneous polynomials; this definition refines the classical construction used by Plücker and later formalized by Hilbert and Noether. Projective varieties are Zariski-closed, hence compact in the Zariski topology; these properties are exploited in treatments by Oscar Zariski and in the foundations of scheme theory by Grothendieck. Over algebraically closed fields like C projective varieties correspond to complex analytic projective sets studied in the work of André Weil and Henri Cartan. Important basic properties include homogeneity under scalar multiplication, behavior under linear projections related to results by Julius Plücker, and the presence of ample divisors as in classical results by Max Noether.
Examples
Classical examples include projective hypersurfaces and complete intersections such as the projective line P^1 and projective plane curves studied by Bernhard Riemann and Felix Klein. Smooth quadric hypersurfaces appear in the work of Arthur Cayley and J. J. Sylvester, while cubic surfaces and K3 surfaces are central in research by C. P. Ramanujam and later by Igor Shafarevich. Flag varieties and Grassmannians are homogeneous projective varieties associated to Lie groups including Élie Cartan’s development of symmetric spaces and later representation-theoretic treatments by Weyl. Famous moduli spaces like the Deligne–Mumford compactification involve projective varieties studied by Pierre Deligne and David Mumford.
Coordinate rings and homogeneous ideals
The homogeneous coordinate ring encodes a projective variety via graded rings used in the approaches of Hilbert and Noether. Hilbert’s basis theorem ensures that homogeneous ideals are finitely generated; this underpins the use of Gröbner bases and computational methods tied to work at institutions such as MIT and Max Planck Institute for Mathematics. The Nullstellensatz in projective form relates radical homogeneous ideals to subvarieties, an extension of results by Emmy Noether and Oscar Zariski. Techniques involving syzygies and free resolutions connect to homological algebra developed by Jean-Pierre Serre and Alexander Grothendieck.
Morphisms and rational maps
Morphisms between projective varieties are given by homogeneous coordinate functions and extend the classical maps considered by Riemann and Klein. Rational maps, indeterminacy loci, and birational equivalence play central roles in birational geometry developed by Francesco Severi and modernized by Shigefumi Mori and the minimal model program associated with Yujiro Kawamata and V. V. Shokurov. Resolution of singularities and birational transformations relate to foundational results by Heisuke Hironaka and techniques used in the proof of theorems by Andrew Wiles in arithmetic geometry contexts.
Dimension, singularities, and smoothness
Dimension theory for projective varieties follows the algebraic definitions formalized by Zariski and later expanded in scheme language by Grothendieck. Singularity theory and the classification of smooth versus singular points connect to work by Oscar Zariski, John Milnor in complex settings, and Kunihiko Kodaira in analytic classifications. Criteria for smoothness use Jacobian matrices and tangent spaces, tools refined in treatments by Serre and in textbooks influenced by the teachings at Harvard University and Cambridge University.
Projective embeddings and line bundles
Projective embeddings arise from very ample line bundles and the global sections studied in the context of the Riemann–Roch theorem by Riemann and extended by Hirzebruch and Atiyah. The concept of ampleness and basepoint-free linear systems is central to embeddings into projective space as developed by Andreotti and Frankel and used in moduli problems by Mumford and Gieseker. Techniques involving cohomology of line bundles and vanishing theorems trace to work by Kodaira, Serre, and Kawamata.
Applications and connections to algebraic geometry
Projective varieties are pervasive across algebraic geometry, arithmetic geometry, and mathematical physics. They underpin the study of curves in number theory researched by Gerd Faltings and Andrew Wiles, inform moduli spaces constructed by Deligne and Mumford, and appear in string-theoretic and mirror symmetry contexts linked to work by Maxim Kontsevich and Cumrun Vafa. Interactions with category theory, derived categories, and homological mirror symmetry involve contributors such as Alexander Grothendieck and Pierre Deligne, while computational and algorithmic aspects connect to projects at Institute for Computational and Experimental Research in Mathematics and Lawrence Berkeley National Laboratory.