Ghillardi4Pi

2022-12-04

Show that the image of ${\mathbb{P}}^{n}×{\mathbb{P}}^{m}$ under the Segre embedding $\psi$ is actually irreducible.

Gwendolyn Case

Expert

So,show the preimage under the Segre map of an algebraic set is an algebraic subset of ${\mathbb{P}}^{n}×{\mathbb{P}}^{m}.$. If $V$ were reducible, we could write $V={V}_{1}\cup {V}_{2},$, where ${V}_{i}\ne V$ are closed. Then
${\mathbb{P}}^{n}×{\mathbb{P}}^{m}={S}^{-1}\left(V\right)={S}^{-1}\left({V}_{1}\right)\bigcup {S}^{-1}\left({V}_{2}\right)$
where ${S}^{-1}\left({V}_{i}\right)$ are closed. Picking ${x}_{i}\in V\setminus {V}_{i}$ we have ${S}^{-1}\left({x}_{i}\right)\cap {S}^{-1}\left({V}_{i}\right)=\mathrm{\varnothing }$ so ${S}^{-1}\left({V}_{i}\right)\subset {S}^{-1}\left(V\right)$ is a strict inclusion, contradicting that ${\mathbb{P}}^{n}×{\mathbb{P}}^{m}$ is irreducible.

Do you have a similar question?