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?



Picard group of hypersurfaces in $\mathbb{P}^r\times\mathbb{P}^s$
Ok, this follows from Lazarsfeld positivity I, Example 3.1.25.


