TITLE: On a Hirzebruch surface
QUESTION [16 upvotes]: I am trying to solve exercise in Huybrechts's book 'Complex geometry'
While solving problems, one problem kept me from going forward.
That is,
The surface $\Sigma_n=\mathbb{P}$ $(\mathcal{O}_ {\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1} (n))$ is n-th Hirzebruch surface.
Show that $\Sigma_n$ is isomorphic to the hypersurface
{${([x_0 , x_1 ],[y_0,y_1,y_2]):x_0^n y_1 - x_1^n y_2 =0}$}$\subset \mathbb{P}^1 \times \mathbb{P}^2$.
How to attack this?
I hope someone shed me a light!

REPLY [2 votes]: The holomorphic section of $\mathcal O_\mathbb{P^1}(n)$ is canonically isomorphic to $\mathbb C[z_0,z_1]_n$ of all homogeneous polynomial of degree n, which is an n+1 dimension vector space. Let $s_0$, $s_1$ be two holomorphic section corresponds to the polynomial $z_{0}^n$ and $z_{1}^n$ respectively. 
Then we have a map $\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}\longrightarrow\mathcal{O}_{\mathbb{P}^1}(n) $ by sending $(f,g)$ to $fs_0+gs_1.$ This map will induces an isomorphism from $\Sigma_n$ to $\mathbb{P}(\mathcal{O}_ {\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1} (n))$.
If you want to write the isomorphism explicitly. Here we define the embbeding $\phi:\mathbb{P}(\mathcal{O}_ {\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1} (n))\longrightarrow \mathbb P^1\times \mathbb P^2$
On $\mathbb{P}(\mathcal{O}_ {\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1} (n))\mid_{U_0}$, this map is give by
$U_{0}\times \mathbb P^{1}\longrightarrow \mathbb P^1\times \mathbb P^2\\
[x_0,x_1]\times[a,b]\mapsto[x_0,x_1]\times[a,b\frac{{x_1}^n}{{x_0}^n},b]$
On $\mathbb{P}(\mathcal{O}_ {\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1} (n))\mid_{U_1}$, this map is give by
$U_{1}\times \mathbb P^{1}\longrightarrow \mathbb P^1\times \mathbb P^2\\
[x_0,x_1]\times[a,b]\mapsto[x_0,x_1]\times[a,b,b\frac{{x_0}^n}{{x_1}^n}]$
With the definition above $\phi$ is well defined and is the embedding we are looking for.