Let
$$
\varphi_{n,m}:\mathbb{P}^{n}\times \mathbb{P}^{m}\rightarrow \mathbb{P}^{nm+n+m}, ((p_{0}:\ldots:p_{n})(q_{0}:\ldots:q_{m}))\mapsto (p_{0}q_{0}:p_{0}q_{1}:\ldots:p_{n}q_{m})
$$
be the Segre embedding. I have to prove that if $X\subseteq \mathbb{P}^{n},Y\subseteq\mathbb{P}^{m}$ are quasi-projective varieties, then $\varphi_{n,m}(X\times Y)$ is a quasi-projective variety. Let $W\subseteq \mathbb{P}^{n},Z\subseteq\mathbb{P}^{m}$ be closed sets and $U\subseteq \mathbb{P}^{n},V\subseteq\mathbb{P}^{m}$ be open sets such that
$$
X=W\cap U,
$$
$$
Y=Z\cap V.
$$
We may assume that $X,Y$ are dense in $W,Z$ respectively. According to this, $X$ (resp. $Y$) is irreducible if and only if $W$ (resp. $Z$) is irreducible.
Let
$$
\pi_{1}: \mathbb{P}^{n}\times \mathbb{P}^{m}\rightarrow\mathbb{P}^{n}
$$
$$
\pi_{2}:\mathbb{P}^{n}\times \mathbb{P}^{m}\rightarrow \mathbb{P}^{m}
$$
be the projections.
I am able to prove that
$$
\varphi_{n,m}(X\times Y)=[\varphi_{n,m}(\pi_{1}^{-1}(W))\cap\varphi_{n,m}(\pi_{2}^{-1}(Z))]\cap[\varphi_{n,m}(\pi_{1}^{-1}(U))\cap\varphi_{n,m}(\pi_{2}^{-1}(V))]
$$
Hence, $\varphi_{n,m}(X\times Y)$ is a quasi-projective set, because the first set is closed and the second is open. Now, How do we know that it is irreducible?
Best Answer
Assume otherwise. Then there exist two algebraic sets $A \cup B=X \times Y$. Now, given any fiber $X_i=\pi_1^{-1}(x_i)=\{x_i\}\times Y$ of a point $x_i\in X$, $X_i$ is isomorphic to $Y$, hence irreducible. Write $X_i=(A\cap X_i)\cup (B \cap X_i)$. Then either $X_i \subset A$ or $X_i \subset B$. Let $X_A, X_B$ denote the set of $x_i$ such that $X_i \subset A, B$ respectively. If we can prove that $X_A, X_B$ are closed, then, since $X = X_A \cup X_B$, either $X_A$ or $X_B =X$. But, $X_A=\cap \pi_1^{-1}((X\times \{y_i\})\cap A)$ where the intersection ranges over all $y_i\in Y$. Thus $X_A$ is the intersection of closed sets, and hence closed.