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(EDIT: I’ve updated this question to feature a new cubic. Sorry for asking so many similar questions. The responses that I’ve received so far have been extremely helpful.)

Consider a cubic in $P^2(\mathbb{C})$ defined by a conic and its tangent. Write a parametrization $\gamma(t)$ such that $t=\infty$ corresponds to the unique singular point, and the sum of the parameters of three collinear points is equal to 0.

I think this problem is related to a question that I posted recently: Unique group law on cubic

I’d like to work with the cubic $F(x,y,z)=(x^2+y^2-z^2)(x-z)$. But I can’t see how to parametrize this curve in $P^2(\mathbb{C})$. I think something like $\gamma(t)=[t^2-1:-2t: t^2+1]$ should parametrize the conic, but I’m not sure how to get the tangent line as well… It seems I’d need more parameters.

I calculated the singular point of this cubic to be $[1: 0: 1]$, which $\gamma(t)$ approaches as $t$ goes to infinity. Am I on to something here?

ArbitraryElement
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  • You keep saying projective curve, so you answered your question. Your parametrization $[1/t^3,1/t^2,1] = [1,t,t^3]$ is correct. Now work out the condition for points with parameters $s,t,u$ to be collinear. – Ted Shifrin Mar 26 '24 at 21:54
  • Thanks for responding. I've updated the question. – ArbitraryElement Mar 28 '24 at 19:27
  • You should not change the question in the middle of discussion. Regarding the new question, "and its tangent" does not make much sense. Do you mean the tangent line at some particular point? And you keep confusing dimensions: It's a curve in $\Bbb P^2$ and a tangent line. The rational parametrization of a conic is not going to help you parametrize this singular cubic. Think about how you parametrized a singular cubic in the previous cases. At what point is the line you took tangent to the conic? – Ted Shifrin Mar 28 '24 at 19:42
  • Unfortunately, that's how the problem was worded to me. The "tangent line at some particular point" is my interpretation. – ArbitraryElement Mar 28 '24 at 19:46
  • That's more accurate than saying "its tangent." At any rate, you have a smooth conic. At what point is its tangent line the line $x-z=0$? – Ted Shifrin Mar 28 '24 at 19:48
  • That would be the point [1 : 0 : 1]. Am I looking for a trigonometric parametrization? – ArbitraryElement Mar 28 '24 at 20:30
  • No. You should proceed as in the other cases, looking at the pencil of lines through the singular point. – Ted Shifrin Mar 28 '24 at 20:33
  • Sorry - I thought I had the idea, but I'm still not understanding. The pencil of lines through the singular point gives me a parametrization of the conic, but I'm not seeing how to parametrize the whole cubic using a single parameter. Would you mind elaborating? – ArbitraryElement Mar 30 '24 at 03:08
  • I spoke too quickly. A reducible curve such as this cannot be rationally parametrized. – Ted Shifrin Mar 30 '24 at 05:18

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