Cayley–Bacharach theorem
Statement about cubic curves in the projective plane / From Wikipedia, the free encyclopedia
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In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states:
- Assume that two cubics C1 and C2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point.
A more intrinsic form of the Cayley–Bacharach theorem reads as follows:
- Every cubic curve C over an algebraically closed field that passes through a given set of eight points P1, ..., P8 also passes through (counting multiplicities) a ninth point P9 which depends only on P1, ..., P8.
A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach.[1]