Picard group and $\mathbb{A}^1$ homotopy

Question

In Morel-Voevodsky $\mathbb{A}^1$-homotopy there is a famous theorem that states $\mathrm{Pic}(X)=[X,\mathbb{P}^{\infty}].$ Can you give me an example of computation of the Picard group of a quasi-projective variety or curve using $\mathbb{A}^1$-homotopy theory that otherwise would be hard to compute? Or for a given variety $X$ first find an $\mathbb{A}^1$-homotopy equivalent variery say $Y$, such that $[Y,\mathbb{P}^{\infty}]=\mathrm{Pic} \ (Y)$ is easier to compute ?