Cayley–Bacharach theorem

In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane $P 2$ . The original form states:

Assume that two cubics

C 1

and

C 2

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

P 1, ..., P 8

also passes through (counting multiplicities) a ninth point

P 9

which depends only on

P 1, ..., P 8

.

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.

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