About: Complete intersection

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In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials: in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V.

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rdf:type	road junction
rdfs:label	Complete intersection (en)
rdfs:comment	In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials: in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V. (en)
dcterms:subject	Commutative algebra Algebraic geometry
Wikipage page ID	3116835 (xsd:integer)
Wikipage revision ID	1068681031 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective space Algebraically closed field Universal coefficient theorem Transversality (mathematics) Complex number General position Elliptic curve Multiset Commutative algebra Tangent space Tuple Algebraic variety Dimension of an algebraic variety Lefschetz hyperplane theorem Radical of an ideal Hypersurface Commutative algebra Algebraic geometry Homogeneous coordinates Torsion-free group Springer Science+Business Media Kunihiko Kodaira Field of fractions Scheme-theoretic intersection Hodge number Twisted cubic Regular sequence (algebra) Homogeneous ideal Homology groups Local complete intersection
Link from a Wikipage to an external page	https://web.archive.org/web/20170815175200/http:/www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0399116-4/S0002-9939-1976-0399116-4.pdf https://www.ams.org/journals/proc/1976-056-01/S0002-9939-1976-0399116-4/S0002-9939-1976-0399116-4.pdf https://www.springer.com/gp/book/9780387977164 http://www.map.mpim-bonn.mpg.de/Complete_intersections
sameAs	Complete intersection Complete intersection Complete intersection
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_book dbt:For dbt:Reflist dbt:Sfn
has abstract	In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials: in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V. Geometrically, each Fi defines a hypersurface; the intersection of these hypersurfaces should be V. The intersection of n − m hypersurfaces will always have dimension at least m, assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to m, with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension n − m ≥ 2. When n − m = 1 then V is automatically a hypersurface and there is nothing to prove. (en)
gold:hypernym	Intersection
prov:wasDerivedFrom	wikipedia-en:Complete_intersection?oldid=1068681031&ns=0
page length (characters) of wiki page	7597 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Complete_intersection
is Link from a Wikipage to another Wikipage of	List of algebraic geometry topics Curve Wiles's proof of Fermat's Last Theorem Combinatorial mirror symmetry Equations defining abelian varieties Calabi–Yau manifold Complete intersection ring Adjunction formula Quintic threefold Fano variety Hilbert series and Hilbert polynomial Primary decomposition Surface (mathematics) Homogeneous coordinate ring Homological mirror symmetry Mixed Hodge structure Regular sequence Picard group Orthogonal group Canonical bundle Rational normal curve Max Noether's theorem on curves Nash blowing-up Scheme-theoretic intersection Glossary of algebraic geometry Surface of general type Twisted cubic
is foaf:primaryTopic of	wikipedia-en:Complete_intersection

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