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I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following:

(i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in affine coordinates by $\phi(x,y)=(x,x^{2}, y)$.

(ii) Find irreducible quadric and cubic hypersurfaces in $\mathbb{P}^{3}$ that contain $\phi(X)$.

I am rather lost on how to approach this. Any help would be appreciated! Thanks very much.

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  • $\begingroup$ Hi! Is this homework? $\endgroup$ – Joachim Apr 21 '14 at 20:46
  • $\begingroup$ Hi! Nope, but I believe it is from a previous homework assignment of a friend of mine. We've both had trouble figuring it out :( $\endgroup$ – user 3462 Apr 21 '14 at 22:32

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