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- Hodge_index_theorem abstract "In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.In a more formal statement, specify that V is a non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersectionwhere d is the degree of V (in that embedding). Let D be the vector space of rational divisor classes on V, up to algebraic equivalence. The dimension of D is finite and is usually denoted by ρ(V). The Hodge index theorem says that the subspace spanned by H in D has a complementary subspace on which the intersection pairing is negative definite. Therefore the signature (often also called index) is (1,ρ(V)-1).The abelian group of divisor classes up to algebraic equivalence is now called the Néron-Severi group; it is known to be a finitely-generated abelian group, and the result is about its tensor product with the rational number field. Therefore ρ(V) is equally the rank of the Néron-Severi group (which can have a non-trivial torsion subgroup, on occasion). This result was proved in the 1930s by W. V. D. Hodge, for varieties over the complex numbers, after it had been a conjecture for some time of the Italian school of algebraic geometry (in particular, Francesco Severi, who in this case showed that ρ < ∞). Hodge's methods were the topological ones brought in by Lefschetz. The result holds over general (algebraically closed) fields.".
- Hodge_index_theorem wikiPageID "1242892".
- Hodge_index_theorem wikiPageRevisionID "592812358".
- Hodge_index_theorem hasPhotoCollection Hodge_index_theorem.
- Hodge_index_theorem subject Category:Algebraic_surfaces.
- Hodge_index_theorem subject Category:Geometry_of_divisors.
- Hodge_index_theorem subject Category:Intersection_theory.
- Hodge_index_theorem subject Category:Theorems_in_algebraic_geometry.
- Hodge_index_theorem type AlgebraicSurfaces.
- Hodge_index_theorem type Artifact100021939.
- Hodge_index_theorem type Object100002684.
- Hodge_index_theorem type PhysicalEntity100001930.
- Hodge_index_theorem type Surface104362025.
- Hodge_index_theorem type Whole100003553.
- Hodge_index_theorem comment "In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V.".
- Hodge_index_theorem label "Hodge index theorem".
- Hodge_index_theorem sameAs m.04lksn.
- Hodge_index_theorem sameAs Q5876058.
- Hodge_index_theorem sameAs Q5876058.
- Hodge_index_theorem sameAs Hodge_index_theorem.
- Hodge_index_theorem wasDerivedFrom Hodge_index_theorem?oldid=592812358.
- Hodge_index_theorem isPrimaryTopicOf Hodge_index_theorem.