About: Projective variety

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In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of . If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

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rdf:type	Thing yago:Abstraction100002137 yago:Assortment108398773 yago:Collection107951464 yago:Group100031264 yago:WikicatAlgebraicVarieties
rdfs:label	Projektive Varietät (de) Varietà proiettiva (it) Variété projective (fr) 射影多様体 (ja) Projective variety (en)
rdfs:comment	In der klassischen algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine projektive Varietät ein geometrisches Objekt, das durch homogene Polynome beschrieben werden kann. (de) En géométrie algébrique, les variétés projectives forment une classe importante de variétés. Elles vérifient des propriétés de compacité et des propriétés de finitude. C'est l'objet central de la géométrie algébrique globale. Sur un corps algébriquement clos, les points d'une variété projective sont les points d'un ensemble algébrique projectif. (fr) Una varietà proiettiva è l'insieme dei punti di uno spazio proiettivo -dimensionale (dove è un campo) che annullano simultaneamente una data famiglia di polinomi omogenei di , ossia Sebbene tale assunzione non sia universalmente accettata, nella letteratura matematica recente si suppone, nella definizione di varietà proiettiva, che essa sia nella topologia di Zariski. Senza tale richiesta si parla invece di insieme algebrico proiettivo. (it) In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of . If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring (en) 代数幾何学において，代数閉体 k 上の射影多様体（しゃえいたようたい，英: projective variety）とは，k 上の（n 次元）射影空間 Pn の部分集合であって，素イデアルを生成する k 係数 n + 1 変数斉次多項式の有限族の零点集合として書けるものをいう．そのようなイデアルは多様体の定義イデアルと呼ばれる．あるいは同じことだが，代数多様体が射影的であるとは，Pn のザリスキ閉として埋め込めるときにいう． 1次元の射影多様体は射影曲線と呼ばれ，2次元だと射影曲面，余次元 1 だと射影超曲面と呼ばれる．射影超曲面は単独の斉次式の零点集合である．射影多様体 X が斉次素イデアル I によって定義されているとき，商環は X の斉次座標環と呼ばれる．やのような基本的な不変量は，この次数環のヒルベルト多項式から読み取ることができる．射影多様体は多くの方法で生じる．それらはであり，荒っぽく言えば「抜けている」点がない．逆は一般には正しくないが，はこの2つの概念の近い関係を記述する．多様体が射影的であることは直線束や因子を調べることによって示される． (ja)
rdfs:seeAlso	Complex projective space
foaf:depiction
dcterms:subject	Algebraic varieties Projective geometry Algebraic geometry
Wikipage page ID	320469 (xsd:integer)
Wikipage revision ID	1094549054 (xsd:integer)
Link from a Wikipage to another Wikipage	Canonical model Prime ideal Meromorphic function Monomial Projectivization Normal variety Algebraic geometry of projective spaces Algebraic group Algebraically closed field Algebraic varieties Holomorphic function Holomorphic vector bundle Join (algebraic geometry) Degree of an algebraic variety Integrally closed domain Jacobian variety Kähler differential Affine scheme Quotient ring Weighted projective space Complex number Analytic space Chow ring General linear group Generalized flag variety Normalization of an algebraic variety Pullback Quasi-projective variety Elliptic curve Geometric genus Geometric invariant theory Grassmannian Ergebnisse der Mathematik und ihrer Grenzgebiete Nilpotent Arithmetic genus Liouville's theorem (complex analysis) Chow's lemma Chow variety Singularity theory Closed immersion Complete variety Complex Lie group Complex manifold Complex projective space Complex torus Function field of an algebraic variety Picard group Polarization of an algebraic form Main theorem of elimination theory Spectrum of a ring Bézout's theorem Adequate equivalence relation Triangular matrices Weil divisor Line bundle Linear system of divisors Poincaré duality Affine variety Algebraic geometry Algebraic geometry and analytic geometry Algebraic number theory Algebraic variety Ample line bundle Projective geometry Noether's normalization lemma Base locus Dimension of an algebraic variety Direct sum Discrete valuation ring Graded ring Hirzebruch–Riemann–Roch theorem Isomorphism Kodaira embedding theorem Kodaira vanishing theorem Lefschetz hyperplane theorem Proper map Proper morphism

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