= Picard group of a smooth affine quadric surface
{c}
{title2=$\operatorname{Pic}(Q\setminus C)\cong\mathbb Z$}
Over an <algebraically closed field>, remove a smooth hyperplane conic $C$ from a <smooth quadric surface> $Q$. The <rulings of a smooth quadric surface> generate $\operatorname{Pic}(Q)=\mathbb Z^2$, and $C$ has class $(1,1)$. <Picard-group localization on a smooth variety> gives the quotient by $\mathbb Z(1,1)$. The restricted ruling bundle $\mathcal O_Q(1,0)$ is a generator, exhibiting a nontrivial <line bundle> on an affine surface.
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