Cayley–Bacharach theorem
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:
A more intrinsic form of the Cayley–Bacharach theorem reads as follows:
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 (1886).
If seven of the points P1, ..., P8 lie on a conic, then the ninth point can be chosen on that conic, since C will always contain the whole conic on account of Bézout's theorem. In other cases, we have the following.
In that case, every cubic through P1, ..., P8 also passes through the intersection of any two different cubics through P1, ..., P8, which has at least nine points (over the algebraic closure) on account of Bézout's theorem. These points cannot be covered by P1, ..., P8 only, which gives us P9.
Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear. Consequently:
On the other hand, assume P1, P2, P3, P4 are collinear and no seven points out of P1, ..., P8 are co-conic. Then no five points of P1, ..., P8 and no three points of P5, P6, P7, P8 are collinear. Since C will always contain the whole line through P1, P2, P3, P4 on account of Bézout's theorem, the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 is isomorphic to the vector space of quadratic homogeneous polynomials that vanish (the affine cones of) P5, P6, P7, P8, which has dimension two.
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