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If $P$ and $Q$ are two sufficiently general points on a cubic surface, the line between them intersects the cubic surface at a unique third point, $f(P,Q)$. This gives a binary operation on (generic) points on a cubic surface.

What identities does this binary operation satisfy?

For instance, it clearly satisfies the identites $f(P,Q) =f(Q,P)$ and $f(P,f(P,Q) ) = Q$.

For an elliptic curve, we can easily find the answer to the analogous question using the fact that $f(P,Q)=P^{-1}Q^{-1}$ in some abelian group. This also tells us the answer for identities with at most three variables - since any three points lie on a plane, the question reduces to the elliptic curve case. But what about four variables? Are there any identities with four or more variables that hold on a cubic surface but are not formally implied by the three variable identities?

Lastly, these identities clearly define a variety in the sense of universal algebra? What properties does this variety have? What does a free algebra on $n$ elements look like?

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11  
There is a book by Manin on, more or less, this subject: Cubic forms. –  Mariano Suárez-Alvarez May 9 '13 at 4:50
1  
(You should get the second edition) –  Mariano Suárez-Alvarez May 9 '13 at 4:56
    
What is f(P,P)? I presume it will be P if you are serious about looking at the identities. Gerhard "Ask Me About System Design" Paseman, 2013.05.08 –  Gerhard Paseman May 9 '13 at 5:31
    
$f(P,P)$ is undefined. On an elliptic curve, it has a definite value, which is usually not $P$, but each point on a cubic surface is on many elliptic curves. –  Will Sawin May 9 '13 at 15:37

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