Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be sufficiently large integers.

Is it true that, for any irreducible hypersurface $X$ of bi-degree $(d,1)$ in $\mathbb{P}^r\times\mathbb{P}^r$, the Picard group $\mathrm{Pic}(X)$ or the divisor class group $\mathrm{Cl}(X)$ equals to $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the Lefschetz holds for such singular hypersurfaces?

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