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
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Hodge index theorem, Wikipedia
Statement sectionQuote, Statement section
The signature (often also called index) is (1,rho(V)-1).
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