Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then,
1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X \otimes_{\mathbb{C}} \mathcal{O}_Y$?
2) Is the ideal sheaf of $X \times_{\mathbb{C}} Y$ in $\mathbb{P}^n \times \mathbb{P}^m$ isomorphic to $\mathcal{I}_{X|\mathbb{P}^n} \otimes_{\mathbb{C}} \mathcal{I}_{Y|\mathbb{P}^m}$?
The first is correct and the second is wrong. For the second, for example take $X=Y=\mathbb{P}^1\subset\mathbb{P}^2$, lines in the plane. Then $I_X\otimes I_Y=\mathcal{O}_{\mathbb{P}^2}(-1)\otimes \mathcal{O}_{\mathbb{P}^2}(-1)$, a line bundle on $\mathbb{P}^2\times\mathbb{P}^2$, so this ideal sheaf defines a divisor. But $\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^2\times\mathbb{P}^2$ is codimension 2 and not a divisor.