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• Projective_variety abstract "In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pn 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. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of Pn. A Zariski open subvariety of a projective variety is called a quasi-projective variety.If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ringis called the homogeneous coordinate ring of X. The ring comes with the Hilbert polynomial P, an important invariant (depending on embedding) of X. The degree of P is the topological dimension r of X and its leading coefficient times r! is the degree of the variety X. The arithmetic genus of X is (−1)r (P(0) − 1) when X is smooth. For example, the homogeneous coordinate ring of Pn is and its Hilbert polynomial is its arithmetic genus is zero.Another important invariant of a projective variety X is the Picard group of X, the set of isomorphism classes of line bundles on X. It is isomorphic to . It is an intrinsic notion (independent of embedding). For example, the Picard group of Pn is isomorphic to Z via the degree map. The kernel of is called the Jacobian variety of X. The Jacobian of a (smooth) curve plays an important role in the study of the curve.The classification program, classical and modern, naturally leads to the construction of moduli of projective varieties. A Hilbert scheme, which is a projective scheme, is used to parametrize closed subschemes of Pn with the prescribed Hilbert polynomial. For example, a Grassmannian is a Hilbert scheme with the specific Hilbert polynomial. The geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.For complex projective varieties, there is a marriage of algebraic and complex-analytic approaches. Chow's theorem says that a subset of the projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. (A corollary of this is that a "compact" complex space admits at most one variety structure.) The GAGA says that the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles.".
• Projective_variety thumbnail Addition_on_cubic_(clean_version).svg?width=300.
• Projective_variety wikiPageExternalLink 216blog.
• Projective_variety wikiPageExternalLink the-hilbert-scheme.
• Projective_variety wikiPageExternalLink math624_08.html.
• Projective_variety wikiPageExternalLink 764ProjectiveVarieties.pdf.
• Projective_variety wikiPageExternalLink ~kollar.
• Projective_variety wikiPageID "320469".
• Projective_variety wikiPageRevisionID "593634330".
• Projective_variety hasPhotoCollection Projective_variety.
• Projective_variety subject Category:Algebraic_geometry.
• Projective_variety subject Category:Algebraic_varieties.
• Projective_variety subject Category:Projective_geometry.
• Projective_variety comment "In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pn 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. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set.".
• Projective_variety label "Projective variety".
• Projective_variety label "Projektive Varietät".
• Projective_variety label "Varietà proiettiva".
• Projective_variety label "Variété algébrique projective".
• Projective_variety sameAs Projektive_Varietät.
• Projective_variety sameAs Variété_algébrique_projective.
• Projective_variety sameAs Varietà_proiettiva.
• Projective_variety sameAs 射影多様体.
• Projective_variety sameAs m.0lkzgql.
• Projective_variety sameAs Q3554818.
• Projective_variety sameAs Q3554818.
• Projective_variety wasDerivedFrom Projective_variety?oldid=593634330.
• Projective_variety depiction Addition_on_cubic_(clean_version).svg.
• Projective_variety isPrimaryTopicOf Projective_variety.