Cubic curves

Dear Professor Kim,

Thank you for your reply. I have another question, this time regarding the group law on a cubic, C. I am happy with showing that the points of C form a group. However I am a little unsure of the ‘simplified’ group law as described in the textbook. I believe that if we take O=(0,1,0) as the identity element we can treat the cubic as an affine curve. However as this corresponds to the ‘line at infinity’ in projective 2-space, what point (x,y) does it correspond to in affine 2-space?
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Reply:

IF C is the curve Y^2Z=X^3+axZ^2+bZ^3, in P^2, then it meets the line Z=0 in the point O=(0:1,0) as explained in the book. So we then express C as a union

C=\{O\}\cup C_0

where C_0=C\cap \{(X:Y:Z) | Z\neq 0\}. We then note that
\{(X:Y:Z) | Z\neq 0\}\simeq A^2 by the map

(X:Y:Z)\mapsto (X/Z,Y/Z).

Then C_0 goes to the curve in A^2 with equation

y^2=x^3+ax+b.

Of course the origin does not lie on this curve, because this affine curve corresponds exactly to the complement of the origin in C.

Does this answer your question? Let me know again if you’re still curious.

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