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One of the most elementary theorems about Picard group is probably $\mathrm{Pic} (X \times \mathbb{A}^n) \cong \mathrm{Pic} X$ and $\mathrm{Pic} (X \times \mathbb{P}^n) \cong \mathrm{Pic} X \times \mathbb{Z}$ (we probably needs need some restriction for $X$ but let's forget about it for now). This looks very similar to the formulas for $\pi_1$ (the fundamental group). So, my question is whether the who has any relationship and whether one can prove those formulas of the Picard groups using some kind of deformation (as in Topology).