show that the products of quasi projective varieties are quasi projective varieties.

use the hint: $(X_1 \setminus X_0) \times (Y_1 \setminus Y_0) = (X_1 \times Y_1) \setminus ((X_0 \times Y_1)\cup (X_1 \times Y_0)). $

attempt: let $X_0 \subset X \subset \mathbb{P}^m$, where $X_0$ is open and $X$ is closed and similarly for $Y_0$. Then $X_0 \times Y_0 = (X \setminus X_1) \times (Y\setminus Y_1) = (X \times Y) \setminus ((X_1 \times Y) \cup (X \setminus Y_1))$.

Could someone please verify this? I am stuck after this. I want to show $X_0 \times Y_0$ is open under the Zariski topology so that it's quasi projective variety

  • 1
    $\begingroup$ So using your argument, can you see that it's enough to show product of projective is projective? $\endgroup$ – Rijul Saini Mar 28 '17 at 6:38
  • $\begingroup$ What are $X_1$ and $Y_1$? $\endgroup$ – Armando j18eos Mar 28 '17 at 11:22
  • $\begingroup$ Hint: Use the Segre embedding. $\endgroup$ – user45878 Apr 6 '17 at 10:33
  • $\begingroup$ How would you do that ? I am given the hint . I don't know how the serge embedding could help $\endgroup$ – Mahidevran Apr 6 '17 at 15:17

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