# Finding a stronger version of Cayley-Bacharach Theorem that applies in the case that the intersection multiplicities are not equal to $1$

Cayley–Bacharach theorem: Assume that two cubics $C_1$ and $C_2$ in the projective plane $\mathbb{P}^2$ meet in nine (different) points. Then every cubic that passes through any eight of the points also passes through the ninth point.

Question (can be found in Gathmann's 2002 Algebraic Geometry notes): Can you find a stronger version of this statement (Cayley-Bacharach Theorem) that applies in the case that the intersection multiplicities in $C_1 \cap C_2$ are not equal to $1$?

I proved Cayley-Bacharach theorem by two different ways and this is an exercise that I found in Gathmann's notes of Algebraic Geometry. He suggests to find a stronger statement for this theorem.

Since this is interesting and could be used to prove the associativity of the addition law on elliptic curves, I research it and the best that I got is the following:

Cubic Cayley-Bacharach Theorem: Let $C_1$ and $C_2$ be cubic curves in $\mathbb{P}^2$ without common components and assume that $C_1$ is smooth. Suppose that $C_3$ is another cubic curve that contains $8$ of the intersection points of $C_1 \cap C_2$ counting multiplicities. Then $C_3$ pass through the ninth point of $C_1 \cap C_2$.

(Found in page 21 of http://goo.gl/QjzQpm )

On wikipedia I found: Every cubic curve $C$ on an algebraically closed field that passes through a given set of eight points $P_1,\ldots,P_8$ also passes through a certain (fixed) ninth point $P_9$, counting multiplicities.

What do you think that is the stronger version that we are supposed to find?

• Please look up Max Noether's theorem, often referred to as $AF+BG$ theorem. – Mohan Jun 26 '15 at 16:45