The Hodge Index Theorem, in algebraic geometry, determines the signature of the intersection pairing on the algebraic curves lying on a smooth algebraic surface, the bilinear form measuring how curves on the surface intersect one another. Named for W. V. D. Hodge, the theorem shows that this intersection form has exactly one positive direction, associated with an ample divisor class, while every direction orthogonal to it is negative, giving the pairing a definite and highly restricted signature. The result is a foundational tool in the study of algebraic surfaces, controlling which divisor classes on a surface can behave like an ample, or positive, class.
Facts
StatementThe signature, also called the index, of the intersection pairing on the algebraic curves of a smooth projective surface V is (1, rho(V) - 1), where rho(V) is the Picard number of V. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Hodge index theorem, Wikipedia
Proved By
Source Hodge index theorem, Wikipedia
Sources
1. Hodge index theorem, Wikipedia
Lead section
Therefore, the signature (often also called index) is (1,ρ(V)-1).
Proved By: William Vallance Douglas Hodge, Lead
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 < ∞).
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