# projective cubic curve to complex projectie space

Suppose we are given the equation $$y^2z = x(x - z)(x - 2z)$$ I would like to define a degree two map $g$ on this curve into complex projective space. I hate to say I am already lost here - how do I start this ?

So an element on complex projective space is an equivalence class of the form $[z,\omega]$ where $(z,\omega) \backsim (\lambda z,\lambda \omega)$ for any nonzero complex number $\lambda$. Also, I can see that the above equation gives a homogeneous polynomial of degree $3$, $$f(x,y,z) = y^2z- x^3- 3x^2z +2z^2x$$ but how can I find the homogeneous coordinates ? My hunch is I should use the identification $\mathbb{C} P \cong S^2$ .. is that a good idea? Any help would be great!

Personally, I find the affine view much more intuitive, and your intuition may be different. I would dehomogenize with respect to $z$, to get the equation $Y^2=X(X-1)(X-2)$, whereupon your map to the complex line is clearly $(X,Y)\mapsto X$. Now rehomogenize to get $(x,y,z)\mapsto(x,z)$. Voilà!