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Projective variety

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in P n {\displaystyle \mathbb {P} ^{n}} of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety.

Variety and scheme structure, Examples and basic invariants & Complex projective varieties

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Overview

Variety and scheme structure

Relation to complete varieties

Examples and basic invariants

Projections

Duality and linear system

Cohomology of coherent sheaves

Smooth projective varieties

Hilbert schemes

Complex projective varieties

Related notions

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Projective variety

Nodes158
Edges157
Triples61
Avg. degree1.99
Density0.012658
Components1

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Projective variety

Top relations

related to Abelian varieties · 16
Projective variety → Another, Examples, For, In, It, Jac, Jacobian, K3, Moreover, Niels Abel, On, Pic, Picard, The, The Jacobian, Varieties
related to GAGA and Chow's theorem · 9
Projective variety → Chow's, Every, If, In, It, Meromorphic, One, The, This
related to Kodaira vanishing · 8
Projective variety → Euler, Kodaira, Kodaira's, Kähler, Serre, Since, The, The Kodaira
related to Degree · 6
Projective variety → Hi's, Indeed, Let, The, There, This
related to The ring of sections · 5
Projective variety → If, Let, Moreover, Proj, Then
related to Relation to complete varieties · 4
Projective variety → By, However, The, There
is a · 3
Projective variety → algebraic variety that is a closed subvariety of a projective space, line bundle of a divisor.Chow's theorem can be shown via Serre's GAGA principle, projective curve if its dimension is one
related to Smooth projective varieties · 3
Projective variety → In, Kähler, Let
related to Homogeneous coordinate ring and Hilbert polynomial · 1
Projective variety → As
related to Related notions · 1
Projective variety → Multi-projective

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Important terminology

projective displaystyle variety mathbb varieties space complex homogeneous dimension theorem closed algebraic ring called degree smooth scheme polynomial one genus

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Projective varietyis aalgebraic variety that is a closed subvariety of a projective space0.90text
Projective varietyis aprojective curve if its dimension is one0.90text
Projective varietyis aline bundle of a divisor.Chow's theorem can be shown via Serre's GAGA principle0.90text
the degreeinstance ofBasic invariants of X0.80text
the dimension can be read off the Hilbert polynomial of this graded ring.Projective varieties arise in many waysinstance ofBasic invariants of X0.80text
Hodge theoryinstance ofThe combination of analytic and algebraic methods for complex projective varieties lead to areas0.80text
the quotient of the general linear group G L ninstance ofFlag varieties0.80text
G L ninstance ofIn marked contrast to affine algebraic groups0.80text
Projective varietyrelated to Abelian varietiesAnother0.60section
Projective varietyrelated to Abelian varietiesPicard0.60section
Projective varietyrelated to Abelian varietiesPic0.60section
Projective varietyrelated to Abelian varietiesIt0.60section

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