I'm reading Hartshorne Ch1 and I'm wondering if there are some connection between projective varieties defined by the same equation in spaces with different dimension.
For example, we know that the surface defined by $(x_0x_1-x_2x_3)$ in $\mathbb{P}^3$ is iso to $\mathbb{P}^1 \times \mathbb{P}^1$. Does the varitey $(x_0x_1-x_2x_3)$ in $\mathbb{P}^4$ or $\forall\ n>3$ has a similar structure? I have difficulties to find out since $\mathbb{P}^n \times \mathbb{A}^1 \cong \mathbb{P}^{n+1}$ and $\mathbb{P}^n \times \mathbb{P}^1 \cong \mathbb{P}^{n+1}$ is incorrect. It seems that there is no iso like $\mathbb{A}^n \times \mathbb{A}^1 \cong \mathbb{A}^{n+1}$ and I don't know how to deal with it.
Any help would be appreciated!