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Given $U,V$ Zariski open sets in $\mathbb{P}^n$ and $\mathbb{P}^m$, under Segre embedding $\phi$, will $\phi (U\times V)$ be open in $\mathbb{P}^{mn+m+n}$? Or if not I guess it will be at least quasiprojective variety? What about $U$ open $V$ closed?

The original question is to prove that given two quasiprojective varieties, then the image under $\phi$ will be quasiprojective variety. I am able to prove it if they are projective, then the image is closed as well, but not sure how to do it for quasiprojective?

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Note that you would not expect $\phi(U\times V)$ to be open, since it's dimension is no more than $nm$, while $\mathbb{P}^{nm+n+m}$ is of greater dimension.

It's true that you will get a quasi-projective variety, and that would indeed be enough for you to finish up your proof. One way of proving this to be true is by using what you have alredy proven: that the Segre embedding maps closed subsets to closed subsets. Now, just look at the complement of your open set:

$\phi((U\times V)^c)=\phi(\mathbb{P}^n\times\mathbb{P}^m) - \phi(U\times V)$

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