I'm taking a beginner course in Elementary Algebraic Geometry (Algebraic Curves) and there is the following exercise. I'm trying to understand the solution. I haven't learned about Elliptic curves, though this seems to be the motivation for this exercise.
Let $C$ be a smooth non-degenerate cubic in $\mathbb{P}_{\mathbb{C}}^{2},$ and let $o$ be an inflection point on $C$. Show that there exists a unique map $$ +: C \times C \rightarrow C,(p, q) \mapsto p+q $$ such that $p+o=p$ for every point $p,$ and $(p+q)+r=o$ if and only if there exists a line $L$ in $\mathbb{P}_{\mathbb{C}}^{2}$ that intersects $C$ in the points $p, q$ and $r$ (taking multiplicities into account: for instance, if $p=q$ then the line should be tangent to $C$ at $p$ ).
Solution: (The uniqueness bit has been omitted. I've mixed in some of my interpretations into the solution, please tell me if there are any errors!) To prove existence, we first define the map $+$, and then show that defining the map the way that we do it gives $+$ the properties we stated in the question.
That is to say, for every point $p$ on $C$, we define $-p$ as the third intersection point of $C$ and the line through $p$ and $o$. By bezout, the point $-p$ is well-defined, and also unique. For all $p$ and $q$ on $C$, using bezout, let $r$ be the third intersection point of $C$ and the line through $p$ and $q$. Again, this is unique and well-defined. And then define $p+q=-r$. We will prove that this definition gives our map the desired properties. Then, $-o=o$ because $o$ is an inflection point of $C$. That is, the line through $o$ and $o$ is the tangent line, which exists with multiplicity at least 3 because $o$ is an inflection point, so using that $o,o,o$ lives on the same line, we get $-o=o$. Also, $p+o=p$ by definition. (Question 1: Is this because by definition, we consider the line passing through $p,o$, and denote its final intersection with $C$ by $r$. Then $p+0 = -r$, but how is $-r$ defined? It is the third intersection point of $C$ and the line through $r,o$. Thus, here it is $p$. So $p+o=p$?)
Moreover, $p+r=o$ if and only if $p, r$ and $o$ are the three intersection points of a line with $C,$ which is equivalent to saying that $p=-r$. (Question 2: Here, I am totally lost in the entire sentence, I don't understand how to get this implication.) Therefore, $(p+q)+r=o$ if and only if $(p+q)=-r$. By definition, $(p+q)=-r$ means that there is a line intersecting $C$ in the points $p, q$ and $r$.