Let $k$ be a field, not neccesarily algebraicly closed though I don't mind.
Let $X\subset\mathbb{A}^{n},\;Y\subset\mathbb{A}^{m}$ be two algebraic sets (not neccesarily irreducible). I am interested in the projective closure $\overline{X\times Y}$ of their product in terms of the projective closures of $X$,$Y$. More specifically, given the chow forms of $\overline{X}$, $\overline{Y}$, can I calculate the chow form of $\overline{X\times Y}$?
I think one can get it in the following way: Let $J(\overline{X},\overline{X})\subset\mathbb{P}^{n+m+1}$ be their join. That is, we embed $\overline{X}$ in the first $m+1$ coordinates of $\mathbb{P}^{m+n+1}$, and we embed $\overline{Y}$ in the remaining $n+1$ coordinates, and then take their join. It is well known that $dimJ(\overline{X},\overline{Y})=dim(\overline{X})+dim(\overline{Y})+1$, so I was thinking that there is a good chance that $J(\overline{X},\overline{Y})\cap\{x_0+...+x_m-x_{m+1}-...-x_{m+n+1}=0\}$ is simply isomorphic to $\overline{X\times Y}$! My logic that a point on this join looks like $(sP:tQ)$ with $P\in\overline{X}$ and $Q\in\overline{Y}$. For fixed $P,Q$, this line will intersect my equation only once, and also these lines can only intersect on $\overline{X}\cup\overline{Y}$. Chow forms of joins and intersections can be calculated. But I'm not sure this construction is meaningful. I hope my question is understood, any reference or hint will be appreciated, likewise any comments on this construction.