Let $L=\{([a_{0}:a_{1}:....:a_{n}],[b_{0}:b_{1}:...:b_{n}]) \in P^{n}* P^{n}: \sum_{i}a_{i}b_{i}=0\}$
Show $P^{n}*P^{n}-L$ is affine.
I don't know where to start. I only know that if we use a Veronese mapping we can show that a projective space $P_{n}$ minus a hyperplane is affine.
Question: "Show $P^{n}*P^{n}-L$ is affine.
I don't know where to start. I only know that if we use a Veronese mapping we can show that a projective space $P_{n}$ minus a hyperplane is affine."
Answer: Let us consider the case $n=1$ and the embedding
$$v: S:=\mathbb{P}^1 \times \mathbb{P}^1 \rightarrow \mathbb{P}^3:=X $$
(defined "pointwise" in the language of Hartshornes book CHI):
$$v((a_0:a_1)(b_0:b_1)):=(a_0b_0: a_0b_1:a_1b_0:a_1b_1).$$
Let $X:=\mathbb{P}^3$ have homogeneous coordinates $z_0,..,z_3$. In the embedding $v$ it follows the subvariety $Z \subseteq S$ defined by the equation $\sum a_ib_i=0$ is defined by the hyperplane $H:=V(l)$ where
$$l:=z_0+z_1+z_2+z_3.$$
It follows $H\cong \mathbb{P}^2$ and $X-H \cong \mathbb{A}^2$. Moreover $S-Z \cong S \cap \mathbb{A}^2 \subseteq \mathbb{A}^2$ is a closed subvariety of $\mathbb{A}^2$. It follows $\mathbb{P}^1 \times \mathbb{P}^1-Z$ is an affine variety.