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Show that every nonsingular quadric in $\Bbb{CP}^2$ is biholomorphic to $\Bbb{CP}^1$. [Hint: Start by thinking about how to parametrize the projective completion of the affine curve $w = z^2$.]

I'm quite new to algebraic geometry so I might be lacking some of the prerequisites here. A nonsingular quadric $Q$ in $\Bbb{CP}^2$ is given by a quadratic polynomial $p$. I think I'm missing the point of the hint, does the projective completion here mean to homogenize the affine curve $w=z^2$ in $\Bbb C^2$? If so, how is it related to the question at hand?

Raul
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    Yes, you should homogenize the equation $w=z^2$ to obtain a projective quadric. The point is that every quadric is isomorphic, so if you can show that this specific one is isomorphic to $\mathbb{CP}^1$ (which is equivalent to the hint), and then show that every quadric is isomorphic to this one by a change of coordinates, then the proof will be done. – D. Brogan Jan 31 '25 at 16:52

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