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It's not to hard to see that the quotient map $\pi\colon \mathrm{C}^{n+1}\backslash \{0\} \to \mathbb CP^n$ is smooth and surjective. Does that imply that it is a submersion as well?

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Generally, a surjective smooth map need not be a submersion. For instance $x\mapsto x^3$ is smooth surjective on the real line but isn't submersive over $0$. However, $\pi$ is submersive, and you can check this in local coordinates, i.e. using charts for the projective space. –  Olivier Bégassat Oct 7 '12 at 19:06
Thanks for your comment! Could you please elaborate a little bit in terms of the charts? I understand that I have to prove that the differential is surjective, so the relation between these concepts is not clear to me. –  user43014 Oct 7 '12 at 22:00

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