Hyperplane section (original) (raw)
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.
| Property | Value |
|---|---|
| dbo: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. (en) |
| dbo:wikiPageID | 3459767 (xsd:integer) |
| dbo:wikiPageLength | 1541 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 705318963 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Projective_space dbr:Scalar_multiplication dbr:Algebraic_set dbr:Mathematical_analysis dbr:Mathematics dbr:Linear_form dbr:Linear_subspace dbr:Algebraic_geometry dbr:Lefschetz_hyperplane_theorem dbr:Lefschetz_pencil dbr:Radon_transform dbr:Intersection_(set_theory) dbr:Irreducible_component dbr:Hyperplane dbc:Algebraic_geometry dbr:Homogeneous_coordinates dbr:Dual_projective_space dbr:Algebraic_subvariety dbr:Bertini's_theorem |
| dbp:wikiPageUsesTemplate | dbt:Hartshorne_AG |
| dct:subject | dbc:Algebraic_geometry |
| gold:hypernym | dbr:Intersection |
| rdf:type | dbo:RoadJunction |
| rdfs: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. Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication. (en) |
| rdfs:label | Hyperplane section (en) |
| owl:sameAs | freebase:Hyperplane section wikidata:Hyperplane section https://global.dbpedia.org/id/4nfmc |
| prov:wasDerivedFrom | wikipedia-en:Hyperplane_section?oldid=705318963&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Hyperplane_section |
| is dbo:wikiPageWikiLink of | dbr:Weil_cohomology_theory dbr:Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne dbr:Stable_vector_bundle dbr:Standard_conjectures_on_algebraic_cycles dbr:Local_cohomology dbr:Lefschetz_hyperplane_theorem dbr:Lefschetz_pencil dbr:Hodge_index_theorem dbr:Séminaire_Nicolas_Bourbaki_(1950–1959) dbr:Theorem_of_Bertini dbr:Max_Noether's_theorem_on_curves dbr:Scheme-theoretic_intersection |
| is foaf:primaryTopic of | wikipedia-en:Hyperplane_section |