Cayley–Bacharach theorem

Remarks

The most quoted case is $d = e = 3$: if two plane cubics meet in nine distinct points, then every cubic through eight of them passes through the ninth. For this case, the cubics through the eight points form a vector space of dimension $10 - 8 = 2$, because the eight points impose independent conditions on cubics. The defining polynomials $F_1, F_2$ of $C_1, C_2$ are independent and lie in this space, so every cubic through the eight points is $\lambda F_1 + \mu F_2$ and vanishes at the ninth point. The general case is usually proved with duality on the zero-dimensional complete intersection $C_1 \cap C_2$. By Bézout's theorem, two curves of degrees $d$ and $e$ without common component meet in $de$ points counted with multiplicity, so the hypothesis says that all intersections are transverse. The cubic case gives a proof of the associativity of the group law on an elliptic curve, and of Pascal's theorem on six points of a conic.

Going deeper

For a student

The theorem says that the intersection points fail to impose independent conditions on curves of degree $d + e - 3$. This is the degree that appears in the adjunction formula for the canonical divisor of a plane curve. It is the source of many classical incidence theorems, such as Pappus's and Pascal's theorems, which come from degenerate cubics made of lines and conics. In modern language it is a consequence of duality for the complete intersection $C_1 \cap C_2$.