Problem 1. Prove that the Segre map $s: \mathbb{P}^n \times \mathbb{P}^m \rightarrow \mathbb{P}^N$ gives an isomorphism of $\mathbb{P}^n \times \mathbb{P}^m$ with a closed subvariety of $P^N$, where $N=n m+n+m$.

Solution: Denote the variables on $\mathbb{P}^N$ by $z_{i j}$ for $0 \leq i \leq n$ and $0 \leq j \leq m$. Set $I=\left(z_{i j} z_{k l}-z_{i l} z_{k j}\right) \subset k\left[z_{i j}\right]$. Then $s\left(\mathbb{P}^n \times \mathbb{P}^m\right) \subset V_{+}(I) \subset \mathbb{P}^N$. Given fixed indexes $i, j$, define $\varphi: V_{+}(I) \cap D_{+}\left(z_{i j}\right) \rightarrow \mathbb{P}^n \times \mathbb{P}^m$ by $\varphi\left(\left(z_{k l}\right)\right)=\left(z_{0 j}: z_{1 j}: \cdots: z_{n j}\right) \times\left(z_{i 0}:\right.$ $\left.\ldots: z_{i m}\right)$. These maps are compatible and define a morphism $\varphi: V_{+}(I) \rightarrow P^n \times \mathbb{P}^m$ which is inverse to the Segre map.

Problem 3. Let $X$ be a pre-variety such that for each pair of points $x, y \in X$ there is an open affine subvariety $U \subset X$ containing both $x$ and $y$.
(a) Show that $X$ is separated.
(b) Show that $\mathbb{P}^n$ has this property.

Solution: (a) Given two morphisms $f, g: Y \rightarrow X$ we show that $D={y \in Y$ $f(y) \neq g(y)}$ is open in $Y$. If $f(y) \neq g(y)$ then take an open affine $U \subset X$ such that $f(y), g(y) \in U$. Then $V=f^{-1}(U) \cap g^{-1}(U) \subset X$ is open and $f$ and $g$ restrict to morphisms $f, g: V \rightarrow U$. Since $U$ is separated, it follows that ${v \in V \mid f(v) \neq g(v)}$ is open in $Y$. Finally, $D$ is the union of such sets.

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