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Let $\pi:X=\mathbb{P}_n\smallsetminus p\rightarrow \mathbb{P}_{n-1}$ be the linear projection. What is the cokernel of the morphism $$ 0\rightarrow \mathcal{O}_{\mathbb{P}_{n-1}}\rightarrow \pi_*(\mathcal{O}_X)? $$

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$X$ is the total space of line bundle $O(1)$ on $P^{n-1}$. Consequently, $$ \pi_*(O_X) = O \oplus O(-1) \oplus O(-2) \oplus \dots $$ and the cokernel is $\oplus_{i\ge 1} O(-i)$.

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  • $\begingroup$ That would give a lot of functions on $\Bbb{P}^n\smallsetminus p$ ... $\endgroup$
    – abx
    Oct 14, 2014 at 5:31
  • $\begingroup$ @abx: of course the sign was wrong, now it is corrected, thanks! $\endgroup$
    – Sasha
    Oct 14, 2014 at 5:55

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