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- Complete_intersection 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 Fi(X0, ..., Xn), 1 ≤ i ≤ n − m,in the homogeneous coordinates Xj, generate all other homogeneous polynomial which vanishes 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.".
- Complete_intersection wikiPageExternalLink Complete_intersections.
- Complete_intersection wikiPageID "3116835".
- Complete_intersection wikiPageRevisionID "602178487".
- Complete_intersection hasPhotoCollection Complete_intersection.
- Complete_intersection subject Category:Algebraic_geometry.
- Complete_intersection subject Category:Commutative_algebra.
- Complete_intersection 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 Fi(X0, ..., Xn), 1 ≤ i ≤ n − m,in the homogeneous coordinates Xj, generate all other homogeneous polynomial which vanishes on V.Geometrically, each Fi defines a hypersurface; the intersection of these hypersurfaces should be V.".
- Complete_intersection label "Complete intersection".
- Complete_intersection sameAs m.08shyw.
- Complete_intersection sameAs Q5156502.
- Complete_intersection sameAs Q5156502.
- Complete_intersection wasDerivedFrom Complete_intersection?oldid=602178487.
- Complete_intersection isPrimaryTopicOf Complete_intersection.