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Let $m,n\geq 0$ be integers and $k$ be an algebraically closed field. $\mathbb P^n \times \mathbb P^m$ is isomorphic to $\mathbb P^{m+n}$ if and only if $m=0$ or $n=0$.

any comments and hints would be highly appreciated.

KReiser
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Bill
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    A clue to the answer can be found here: https://math.stackexchange.com/questions/983568/how-to-show-p1-times-p1-as-projective-variety-by-segre-embedding-is-not-is – Ruben Jul 21 '19 at 19:11
  • If you don't mind working over the complex numbers in the analytic topology, it's pretty easy to see this by computing the topological cohomology. – Tabes Bridges Jul 21 '19 at 20:43
  • Dear @KReiser: I have some missunderstanding with your answer, first line of your answer is a propositon or can you explain it? I just have some basic knowledge about projectve varieties. – Bill Jul 21 '19 at 21:20
  • Dear @Tabes Bridges: I need it to be solved as a variety. Thank you. – Bill Jul 21 '19 at 21:21
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    This is a basic fact about projective space - see for instance Hartshorne theorem I.7.2. – KReiser Jul 22 '19 at 08:11

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