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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 has a one-dimensional subspace on which it is positive definite , 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 intersection
where d is the degree of V. 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. 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 is -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 is equally the rank of the Néron-Severi group.