About: Hyperplane section

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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.

Attributes	Values
rdf:type	road junction
rdfs:label	Hyperplane section (en)
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)
dcterms:subject	Algebraic geometry
Wikipage page ID	3459767 (xsd:integer)
Wikipage revision ID	705318963 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective space Scalar multiplication Algebraic set Mathematical analysis Mathematics Linear form Linear subspace Algebraic geometry Lefschetz hyperplane theorem Lefschetz pencil Radon transform Intersection (set theory) Irreducible component Hyperplane Algebraic geometry Homogeneous coordinates Dual projective space Algebraic subvariety Bertini's theorem
sameAs	Hyperplane section Hyperplane section Hyperplane section
dbp:wikiPageUsesTemplate	dbt:Hartshorne_AG
has 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)
gold:hypernym	Intersection
prov:wasDerivedFrom	wikipedia-en:Hyperplane_section?oldid=705318963&ns=0
page length (characters) of wiki page	1541 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Hyperplane_section
is Link from a Wikipage to another Wikipage of	Weil cohomology theory Decomposition theorem of Beilinson, Bernstein and Deligne Stable vector bundle Standard conjectures on algebraic cycles Local cohomology Lefschetz hyperplane theorem Lefschetz pencil Hodge index theorem Séminaire Nicolas Bourbaki (1950–1959) Theorem of Bertini Max Noether's theorem on curves Scheme-theoretic intersection
is foaf:primaryTopic of	wikipedia-en:Hyperplane_section

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