Lemma 33.30.5. Let $k$ be a field. Let $X$ be a normal variety over $k$. Let $U \subset \mathbf{A}^ n_ k$ be an open subscheme with $k$-rational points $p, q \in U(k)$. For every invertible module $\mathcal{L}$ on $X \times _{\mathop{\mathrm{Spec}}(k)} U$ the restrictions $\mathcal{L}|_{X \times p}$ and $\mathcal{L}|_{X \times q}$ are isomorphic.
Proof. The fibres of $X \times _{\mathop{\mathrm{Spec}}(k)} U \to X$ are open subschemes of affine $n$-space over fields. Hence these fibres have trivial Picard groups by Divisors, Lemma 31.28.4. Applying Divisors, Lemma 31.28.1 we see that $\mathcal{L}$ is the pullback of an invertible module $\mathcal{N}$ on $X$. $\square$
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