# Find an explicit isomorphism from a curve of genre zero to the Riemann sphere

I can't figure out this exercise: i have this singular curve in $\mathbb{P}^2\mathbb{C}$ given by $\{[X,Y,Z]\in \mathbb{P}^2\mathbb{C}:X^2Y^2+Y^2Z^2+X^2Z^2=0\}$, I have shown that its desingularization is a curve of genus zero, but now I must find an explicit isomorphism with the Riemann sphere. I don't know how to do it, could anybody help me with that? thank you

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The word in English is "genus", not "genre". The plural is "genera". –  Zhen Lin Jan 24 '12 at 18:39
This sounds like homework, so here are some hints. (1) There is an easy isomomorphism between this curve and $U^2+V^2+W^2=0$ (also in $\mathbb{P}^2$), do you see it? (2) $U^2+V^2+W^2$ is a conic and can be approached as described at en.wikipedia.org/wiki/Algebraic_curve#Rational_curves –  David Speyer Jan 25 '12 at 16:20
no, i can't see the isomorphism between the curve and $X^2 + Y^2 + Z^2=0$ because i thought $X \mapsto XY, Y \mapsto YZ, Z \mapsto XZ$ but it doesen't seem to be working... –  tigu Jan 28 '12 at 13:13