Cayley–Bacharach theorem

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.

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.

According to Bézout's theorem, two different cubic curves over an algebraically closed field which have no common irreducible component meet in exactly nine points (counted with multiplicity).

A second application is Pappus's hexagon theorem, similar to the above, but the six points are on two lines instead of on a conic.

Finally, a third case is found for proving the associativity of elliptic curve point addition.

Formally, first recall that given two curves of degree d, they define a pencil (one-parameter linear system) of degree d curves by taking projective linear combinations of the defining equations; this corresponds to two points determining a projective line in the parameter space of curves, which is simply projective space.

More concretely, because the vector space of homogeneous polynomials P(x, y, z) of degree three in three variables x, y, z has dimension 10, the system of cubic curves passing through eight (different) points is parametrized by a vector space of dimension ≥ 2 (the vanishing of the polynomial at one point imposes a single linear condition).