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computations of picard groups

\({\mathbf{A}}^n\)

To show \(\operatorname{Pic}({\mathbf{A}}^n) = 1\):

\({\mathbf{P}}^n\)

Use smoothness and then toric geometry to compute the class group: \(\operatorname{Pic}({\mathbf{P}}^n) = {\operatorname{CH}}^1({\mathbf{P}}^n) \cong {\mathbf{Z}}\).

Smooth hypersurfaces in \({\mathbf{P}}^n\)

Identify \(\operatorname{Pic}(X) = {\operatorname{CH}}^1(X)\) and use the Chow exact sequence to get \(\operatorname{Pic}(X) \cong C_d\).