The Cayley-Bacharach Theorem states that if two cubic curves in the plane intersect in exactly nine points, then any other cubic curve passing through eight of those nine points automatically passes through the ninth as well. Named for Arthur Cayley and Isaak Bacharach, it is a classical result of algebraic geometry with many variants relating the points where families of curves intersect, and it underlies facts such as the associativity of the elliptic curve group law.
Facts
StatementIf two cubics in the projective plane meet in nine different points, every cubic through any eight of them also passes through the ninth. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Wolfram MathWorld: Cayley-Bacharach Theorem
Proved By
Sources
1. Cayley-Bacharach theorem (Wikipedia)
IntroductionQuote, Introduction
Then every cubic that passes through any eight of the points also passes through the ninth point.
Wolfram MathWorld: Cayley-Bacharach Theorem
Wolfram MathWorldIn Branch: Algebraic Geometry, Subject classificationsQuote, In Branch: Algebraic Geometry, Subject classifications
Algebra Algebraic Geometry
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