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!

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!

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

{${([x_0 , x_1 ],[y_0,y_1,y_2]):{x_0}^2 y_1 - {x_1}^n y_2 =0}$}$\subset \mathbb{P}^1 \times \mathbb{P}^2$.

How to attack this?

I hope someone shed me a light!

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!