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Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be integers with $r\geq10, s\geq3$, let $x_0,\dots, x_r, y_0,\dots, y_s$ be coordinates of $\mathbb{P}^{r}\times\mathbb{P}^s$. Let $f(x_0,\dots,x_r,y_0,\dots, y_s)$ be an irreducible homogeneous polynomial of bidegree $(d,1)$, where $d$ is a positive integer. (Namely, $f$ is a linear combination of terms $H(x_0,\dots, x_r)y_i$, where $H$ is a homogeneous polynomial of degree $d$. )

Does the hypersurface of bi degree $(d,1)$ has Picard group (or even the weaker notion of divisor class group) $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the Lefschetz holds for such singular hypersurfaces?

[If we delete the canonical generators inherited from the ambient product projective space, it is equivalent to ask, if the affine complement has trivial Picard (or is a UFD).]

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?

Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be integers with $r\geq10, s\geq3$, let $x_0,\dots, x_r, y_0,\dots, y_s$ be coordinates of $\mathbb{P}^{r}\times\mathbb{P}^s$. Let $f(x_0,\dots,x_r,y_0,\dots, y_s)$ be an irreducible homogeneous polynomial of bidegree $(d,1)$, where $d$ is a positive integer. (Namely, $f$ is a linear combination of terms $H(x_0,\dots, x_r)y_i$, where $H$ is a homogeneous polynomial of degree $d$. )

Does the hypersurface of bi degree $(d,1)$ has Picard group (or even the weaker notion of divisor class group) $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the a Lefschetz holds for such singular hypersurfaces?

[If we delete the canonical generators inherited from the ambient product projective space, it is equivalent to ask, if the affine complement has trivial Picard (or is a UFD).]

Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be integers with $r\geq10, s\geq3$, let $x_0,\dots, x_r, y_0,\dots, y_s$ be coordinates of $\mathbb{P}^{r}\times\mathbb{P}^s$. Let $f(x_0,\dots,x_r,y_0,\dots, y_s)$ be an irreducible homogeneous polynomial of bidegree $(d,1)$, where $d$ is a positive integer. (Namely, $f$ is a linear combination of terms $H(x_0,\dots, x_r)y_i$, where $H$ is a homogeneous polynomial of degree $d$. )

Does the hypersurface of bi degree $(d,1)$ has Picard group (or even the weaker notion of divisor class group) $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the Lefschetz holds for such singular hypersurfaces?

[If we delete the canonical generators inherited from the ambient product projective space, it is equivalent to ask, if the affine complement has trivial Picard (or is a UFD).]

Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be integers with $r\geq10, s\geq3$, let $x_0,\dots, x_r, y_0,\dots, y_s$ be coordinates of $\mathbb{P}^{r}\times\mathbb{P}^s$. Let $f(x_0,\dots,x_r,y_0,\dots, y_s)$ be an irreducible homogeneous polynomial of bidegree $(d,1)$, where $d$ is a positive integer. (Namely, $f$ is a linear combination of terms $H(x_0,\dots, x_r)y_i$, where $H$ is a homogeneous polynomial of degree $d$.

Does the hypersurface of bi degree $(d,1)$ has Picard group $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the a Lefschetz holds for such singular hypersurfaces?

[If we delete the canonical generators inherited from the ambient product projective space, it is equivalent to ask, if the affine complement has trivial Picard (or is a UFD).]

Let $k$ be an algebraically closed field, say $k=\mathbb{C}$. Let $r,s$ be integers with $r\geq10, s\geq3$, let $x_0,\dots, x_r, y_0,\dots, y_s$ be coordinates of $\mathbb{P}^{r}\times\mathbb{P}^s$. Let $f(x_0,\dots,x_r,y_0,\dots, y_s)$ be an irreducible homogeneous polynomial of bidegree $(d,1)$, where $d$ is a positive integer. (Namely, $f$ is a linear combination of terms $H(x_0,\dots, x_r)y_i$, where $H$ is a homogeneous polynomial of degree $d$. )

Does the hypersurface of bi degree $(d,1)$ has Picard group (or even the weaker notion of divisor class group) $\mathbb{Z}\oplus\mathbb{Z}$? I am not sure if the a Lefschetz holds for such singular hypersurfaces?

[If we delete the canonical generators inherited from the ambient product projective space, it is equivalent to ask, if the affine complement has trivial Picard (or is a UFD).]