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: 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. 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. If no seven points out of P1, ..., P8 are co-conic, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 (with multiplicity for double points) has dimension two. 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: If no seven points out of P1, ..., P8 lie on a non-degenerate conic, and no four points out of P1, ..., P8 lie on a line, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) P1, ..., P8 has dimension two. On the other hand, assume P1, P2, P3, P4 are collinear and no seven points out of P1, .