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- Hyperplane_section abstract "In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace. Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication.From a geometrical point of view, the most interesting case is when X is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the Lefschetz pencil.".
- Hyperplane_section wikiPageID "3459767".
- Hyperplane_section wikiPageLength "1541".
- Hyperplane_section wikiPageOutDegree "19".
- Hyperplane_section wikiPageRevisionID "705318963".
- Hyperplane_section wikiPageWikiLink Algebraic_geometry.
- Hyperplane_section wikiPageWikiLink Algebraic_variety.
- Hyperplane_section wikiPageWikiLink Category:Algebraic_geometry.
- Hyperplane_section wikiPageWikiLink Duality_(projective_geometry).
- Hyperplane_section wikiPageWikiLink Homogeneous_coordinates.
- Hyperplane_section wikiPageWikiLink Hyperplane.
- Hyperplane_section wikiPageWikiLink Intersection_(set_theory).
- Hyperplane_section wikiPageWikiLink Irreducible_component.
- Hyperplane_section wikiPageWikiLink Lefschetz_hyperplane_theorem.
- Hyperplane_section wikiPageWikiLink Lefschetz_pencil.
- Hyperplane_section wikiPageWikiLink Linear_form.
- Hyperplane_section wikiPageWikiLink Linear_subspace.
- Hyperplane_section wikiPageWikiLink Mathematical_analysis.
- Hyperplane_section wikiPageWikiLink Mathematics.
- Hyperplane_section wikiPageWikiLink Projective_space.
- Hyperplane_section wikiPageWikiLink Radon_transform.
- Hyperplane_section wikiPageWikiLink Scalar_multiplication.
- Hyperplane_section wikiPageWikiLink Theorem_of_Bertini.
- Hyperplane_section wikiPageWikiLinkText "hyperplane section".
- Hyperplane_section wikiPageUsesTemplate Template:Hartshorne_AG.
- Hyperplane_section subject Category:Algebraic_geometry.
- Hyperplane_section hypernym Intersection.
- Hyperplane_section type RoadJunction.
- Hyperplane_section comment "In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace.".
- Hyperplane_section label "Hyperplane section".
- Hyperplane_section sameAs Q5958484.
- Hyperplane_section sameAs m.09dm1v.
- Hyperplane_section sameAs Q5958484.
- Hyperplane_section wasDerivedFrom Hyperplane_section?oldid=705318963.
- Hyperplane_section isPrimaryTopicOf Hyperplane_section.