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In the following link : https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/torsion.pdf , pages : 3 and 4, we find the following paragraph :

They provide a map $ \phi : Y \to \mathbb{P}^4 $ which for a generic choice of the $ \phi_i $ 's satisfies the following properties :

  • $ \phi $ is generically of degree $ 1 $ onto its image, which is a hypersurface $ X_0 \subset \mathbb{P}^4 $ of degree : $ p^3 s = D $

  • $ \phi $ is two-to-one generically over a surface in $ X_0 $, three-to-one generically over a curve in $ X_0 $, at most finitely many points of $ X_0 $ have more than $ 3 $ preimages, and no points have more than $ 4 $ preimages

Could you explain to me how do we do this calculus ? How do we obtain $ 1-1 $ , $ 2-1 $ and $ 3-1 $ correspondances in this operation ?

In which course can we find about this subject please ?

Thanks in advance for your help.

YoYo
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  • I wrote about this counterexample in another answer; the reference I found for this generic projection statement is "Generic projections of algebraic varieties" by Roberts. – Takumi Murayama Sep 04 '17 at 14:11
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    The statement you need for the counterexample is in the first paragraph in the Roberts paper; the most relevant section is §5, especially Theorem 1 and the proof that follows. You will probably need the material from previous sections, though. I don't have the time to write an answer now; in the meantime, have you tried reading the paper? – Takumi Murayama Sep 04 '17 at 14:34
  • Thank you Takumi Murayama. :-) Yes, I'm trying to read it just now. Thank you. I'll be glad if you write me a detailed answer to this subject here if you have enough time. :-) Thank you? :-) – YoYo Sep 04 '17 at 15:15

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