A double cover of $\mathbb{P}^n \times \mathbb{P}^n$

Question

Let $X$ be a double cover of $\mathbb{P}^n \times \mathbb{P}^n$ branching along bi-degree $(d,e)$ hypersurface. Is it possible to write $X$ as a complete intersection of hypersurfaces in some weighted projective space (for certain $d,e$)?

This is motivated by the fact that a double cover of $\mathbb{P}^n$ branching along degree $2d$ hypersurface can be written as a hypersurface in $\mathbb{P}^n_{(1^{n+1},d)}$.

Answer 1

The answer is no. The reason is that the Picard group of a complete intersection of dimension $\ge 3$ in a weighted projective space is of rank 1 and the Picard group of the double cover is of rank $\ge 2$.