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In Vakil’s FOAG, exercise 15.4.M, there is a remark:

15.4.M. EXERCISE: A TORSION PICARD GROUP. Suppose that $Y$ is a hypersurface in $\Bbb{P}^n_k$ corresponding to an irreducible degree d polynomial. Show that $Pic(\Bbb{P}^n_k-Y) = Z/(d)$. (For differential geometers: this is related to the fact that $π_1(\Bbb{P}^n_k - Y)= Z/(d)$.)

Is there any reference for the related differential geometry fact? and what is the full statement for it? Is the topology still using the Zariski topology? If not, should the field $k$ be the $\Bbb{C}$?

Thank you in advance.

onRiv
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    Got it from https://people.math.wisc.edu/~lmaxim/CIMPA-Lectures.pdf . It’s a result about the complex topology $\Bbb{C}\Bbb{P}^n$ – onRiv Jul 13 '23 at 05:42

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