# Help understanding geometric proof (using Cayley - Bacharach) that group law on elliptic curves is associative

I have a specific question about a geometric proof that the group law on elliptic curves is associative, as carried out in Fulton's Algebraic Curves (Section 5.6) or here.

The main thrust is to begin with a curve $$\mathcal{C}$$, and then to find two sets of 9 points (with 8 points in common) on 3 lines on this curve $$\mathcal{C}$$, to posit that each of these sets of points determines some elliptic curve that is not $$\mathcal{C}$$ (let us call these curves $$\mathcal{C}'$$ and $$\mathcal{C}''$$), and then to apply the Cayley-Bacharach Theorem to show that the 9th point on both $$\mathcal{C}'$$ and $$\mathcal{C}''$$ must be the same.

What I do not understand - surely Cayley - Bacharach applies only if $$\mathcal{C}'$$ and $$\mathcal{C}''$$ are distinct curves, and I do not understand why we are able to say that there MUST be another curve going through 9 points of $$\mathcal{C}$$ that is not equal to $$\mathcal{C}$$. Also, how is the proof valid once we recognize that the "different" curves $$\mathcal{C}'$$ and $$\mathcal{C}''$$, for which the only things we know is that they have the same nine points, share these same nine points in common?

I will try to isolate the idea of the proof of the associativity, hope this answers the unclear points. First, we are doing the following, when we use the Cayley-Bacharach link in the OP.

The definition of the sum:

We start with $$O$$, fixed (rational) point on some fixed cubic curve $$E\subset \Bbb P^2\ ,$$ (all spaces defined over a fixed field,) then consider two other points, $$P,Q$$, and use a specific receipt to define the point $$P+Q$$. The notations in loc. cit are rather irritating, and i will never use something like $$PQ$$ (for a point). The point P+Q is defined uniquely as in the "picture":

 O
|
|
P+Q
|
|
|
*---- P------Q


Here, points in triple joined in the picture through a line correspond to points on a line in the geometry (of the affine space where the elliptic curve also lives in). Now consider a further point $$R$$, we want to show: $$(P+Q)+R = P+(Q+R)\ .$$ This means the equality of the following points "in the middle" X and X' of the diagrams:

   O
|
|
P+Q --- X --------- R
|                  |
|                  |
|                  |
A ---- P --------- Q


and

   O ----- Q+R -------- B
|           |
|           |
X' -------- R
|           |
|           |
|           |
P --------- Q


(Above, starting from X, and respectively X', we have to intersect the lines OX and OX' with the curve, the "third point" is then $$(P+Q)+R$$, and respectively $$P+(Q+R)$$.)

So it is natural to show that both diagrams fit in the same picture:

   O ----- Q+R -------- B ------- Line L3
|        |           |
|        |           |
P+Q ----- * --------- R ------- Line L2
|        |           |
|        |           |
|        |           |
A ------ P --------- Q ------- Line L1
|        |           |
|        |           |
|        |           |
Line     Line        Line
M1       M2          M3


The question is explicitly, if the points * = X and respectively * = X', constructed as follows starting with the eight points "on the margin", $$A,P,A,R,B,Q+R,O,P+Q$$ do coincide:

• Consider the lines $$L_1,L_2,L_3$$, then on $$L_1,L_3$$ we already have by construction three points, let $$X$$ be the third point on $$L_2$$. Let $$l_1,l_2,l_3$$ be degree one homogeneous polynomials, so that the equations $$l_1=0$$, $$l_2=0$$, $$l_3=0$$, describe the lines $$L_1,L_2,L_3$$. Then the degree three polynomial $$l_1l_2l_3$$ defines a (degenerated) cubic.
• Consider the lines $$M_1,M_2,M_3$$, then on $$M_1,M_3$$ we already have by construction three points, let $$X'$$ be the third point on $$M_2$$.

The proof forgets now everything about $$X,X'$$, introduces a new point, $$Y$$, defined as the intersection of the lines $$L_2$$ and $$M_2$$. (A priori, this point may or may not lie on the elliptic curve. In the end, all three points $$X,X',Y$$ coincide.) We are now in the position now to apply in the generic case (eight distinct points) the theorem Cayley-Bacharach for the (degenerated) cubic curves \begin{aligned} C_l &:& l_1l_2l_3&=0\ ,\\ C_m &:& m_1m_2m_3&=0\ , \end{aligned} and the given elliptic curve $$E$$.The $$8+1$$ points are $$A,P,A,R,B,Q+R,O,P+Q$$ plus $$Y$$. It follows, that $$Y$$ is also on the cubic $$E$$. We get by construction $$X=Y=X'$$, the relation we wanted.

In case some of the points coincide, we have to use multiplicities, this leads to the solution from [Fulton, Algebraic Curves], a sort of intersection number (as part of an intersection theory) is needed.