About: Degree of an algebraic variety

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In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the varietywith n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem).

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rdfs:label	Degree of an algebraic variety (en)
rdfs:comment	In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the varietywith n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem). (en)
dcterms:subject	Algebraic varieties
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Wikipage revision ID	1072381852 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective space Multiplicity (mathematics) Monomial Algebraic set Algebraically closed field Algebraic varieties Invertible sheaf Mathematics Chow ring General position Ideal (ring theory) Bézout's theorem Total degree Line bundle Linear subspace Linear system of divisors Affine variety Dimension of an algebraic variety Projective line Projective variety Gröbner basis Hyperplane Hypersurface Chern class Codimension Cohomology ring Homogeneous coordinate ring Homogeneous polynomial Dimension Rational normal curve Canonical line bundle Hilbert series
sameAs	Degree of an algebraic variety Degree of an algebraic variety Degree of an algebraic variety Degree of an algebraic variety Degree of an algebraic variety
dbp:wikiPageUsesTemplate	dbt:Math dbt:More_citations_needed dbt:Reflist dbt:Short_description dbt:Slink
has abstract	In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the varietywith n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem). The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space. The degree of a hypersurface is equal to the total degree of its defining equation. A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces. The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations. (en)
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is Link from a Wikipage to another Wikipage of	Deformation (mathematics) Glossary of classical algebraic geometry Bézout's theorem Algebraic curve Algebraic geometry Ample line bundle Degree Hilbert series and Hilbert polynomial Projective variety Rational function Jean-Pierre Demailly Hodge index theorem Rational normal curve Twisted cubic Degree (algebraic geometry) Degree of a projective variety
is Wikipage redirect of	Degree (algebraic geometry) Degree of a projective variety
is Wikipage disambiguates of	Degree
is foaf:primaryTopic of	wikipedia-en:Degree_of_an_algebraic_variety

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