This is related to the $P^1\times P^1$ image through segre embedding birational to $P^2$(Hartshorne I 4.5). I did a quite convoluted way by identifying the $A^1\times A^1\subset P^1\times P^1$ as quasi-projective variety isomorphic to $A^2\subset Im(Segre)$ where $A^2$ is treated as quasi-projective variety sitting inside $Im(Segre)$ and $Segre$ means the segre morphism$:P^1\times P^1\to P^3$.
Consider affine varieties $X\subset A^n\subset P^n,Y\subset A^m \subset P^m$. Through segre embedding, I have $X\times Y$ identified as quasi projective variety of $P^N$ for some $N$. Now $X\times Y$ has affine variety product structure and also has quasi projective variety structure in $P^n\times P^m$ through segre embedding. I have two different product variety structure. I would like to see that affine variety products can be embedded into product of projectives(i.e. Compatibility needs to be checked).
So I have $A^1\times A^1\subset P^1\times P^1$ open through segre embedding. Now I want to say $A^1\times A^1$ has inherent affine variety product structure (i.e. $A^1\times A^1\cong A^2\subset P^2$ where first iso requires product of affine varieties).
Q1: Is this generally true? There is no guarantee that segre embedding identifies affine varieties but projective/quasi-projectives.
Q2: Is the image of affine under segre image affine? The affines might be sitting inside some quasi-projectives and the image of affines might not be affine.
Q3: If image of affine is affine, then is this compatible with projective variety product structure through segre embedding?(i.e. $A^1\times A^1\cong A^2$ as affine product whereas I also have $A^1\times A^1\cong A^2\subset Im(Segre))$