Group law on cubics

Question

I have a question about the group law on cubics.

I know the definition of the group law on cubics. However, I don't know how to define $P+Q$ in this case. A point $P$ is on the cubic $C$ and it is in the finite plane. $L$ is a tangent line at $P$. Suppose $P$ is not a flex point. Then the line $L$ intersects another point $Q$ on the cubic $C$.

Condiser $P+Q$.

The line $\overline{PQ}\ $ intersects $C$ three times, counting multiplicity. So, there is no third intersection point. I just want to know in this case how to define $P+Q$

Answer 1

I would have liked to write this as a comment, but I could not do it because I do not have enough "reputation".

The answer to your question is in section $5.6$ of Fulton's book on algebraic curves, freely available here: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

Answer 2

The group law on elliptic curves is simply defined as three points have sum zero if and only if they are collinear. In the case of tangent the point of tangency counts twice so you have $$P+P+Q=0$$ which is usual to present as $$2P+Q=0$$ In order to have clear what $P+Q$ is, you just act as for algebra and you get $$P+Q=-P$$