Show that the image of P^n xx P^m under the Segre embedding psi is actually irreducible.



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Show that the image of P n × P m under the Segre embedding ψ is actually irreducible.

Answer & Explanation

Gwendolyn Case

Gwendolyn Case

Beginner2022-12-05Added 7 answers

So,show the preimage under the Segre map of an algebraic set is an algebraic subset of P n × P m .. If V were reducible, we could write V = V 1 V 2 ,, where V i V are closed. Then
P n × P m = S 1 ( V ) = S 1 ( V 1 ) S 1 ( V 2 )
where S 1 ( V i ) are closed. Picking x i V V i we have S 1 ( x i ) S 1 ( V i ) = so S 1 ( V i ) S 1 ( V ) is a strict inclusion, contradicting that P n × P m is irreducible.

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