TITLE: A question of Hirzebruch surface $ \mathbb{P}(\mathcal{O}(1) \bigoplus \mathcal{O})$
QUESTION [9 upvotes]: My question comes from my professor. I try my best to understand what the question means, but  it doesn’t work! I even cannot understand the question meaning!   I think I need some hints to answer the question.
We can find the definition of Hirzebruch surface  in [Fulton ” Introduction Toric varieties” page 8].
Which introduce the construction of  Hirzebruch surface $ \mathbb{F}_a $ with $ a \in \mathbb{N}$. And we know that, in fact, $ \mathbb{F}_a $ is a projective bundle $ \pi:  \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(a) \bigoplus \mathcal{O}_{\mathbb{P}^1}) \to \mathbb{P}^1$.
Now, let us consider the case of $a = 1. \  i.e. \ \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(1) \bigoplus \mathcal{O}_{\mathbb{P}^1})$.
In [Fulton], we construct $ \mathbb{F}_1 $ from the toric variety point of view.
My main Question:

But if we just regard $ \mathbb{F}_1$ $\cong$
  $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(1) \bigoplus$
  $\mathcal{O}_{\mathbb{P}^1})$ as a scheme, how do I write down all
  affine open pieces and identifications between any two affine open
  piece which is "compatible" with the toric construction )

What is the meaning of “Compatible”? I have no sense about this important key word.
I follow the definition of $\mathbb{P} (\mathcal{E})$ in [Hartshorne]:
Let $\mathcal{E} = \mathcal{O}_{\mathbb{P}^1}(1) \bigoplus \mathcal{O}_{\mathbb{P}^1}$ be a locally free sheaf on $\mathbb{P}^1$.Recall that in  $\mathbb{P}^1$, there are two standard affine open subsets $D_{+}(x_0) \cong  Spec \ \mathbb{C}[x_0,x_1]_{(x_0)}$ and $D_{+}(x_1) \cong  Spec\ \mathbb{C}[x_0,x_1]_{(x_1)}$
So we have   $\mathcal{E} | _ {D_{+}(x_0)}$   =  $(\mathcal{O}_{\mathbb{P}^1}(1) \bigoplus \mathcal{O}_{\mathbb{P}^1})|_{D_{+}(x_0)}$  $\cong$ $\widetilde{S(1)_{(x_0)}}\bigoplus \widetilde{S_{(x_0)}}$, where $S =  \mathbb{C}[x_0,x_1]$.
We have $\mathcal{E} | _ {D_{+}(x_0)}( D_{+}(x_0)) = x_0 \mathbb{C}[x_0,x_1]_{(x_0)} \bigoplus  \mathbb{C}[x_0,x_1]_{(x_0)}$
Let $ M := x_0 \mathbb{C}[x_0,x_1]_{(x_0)} \bigoplus  \mathbb{C}[x_0,x_1]_{(x_0)}$, then $ M $ is a free $\mathbb{C}[x_0,x_1]_{(x_0)}-module$ of rank 2.
Consider the symmetric algebra of $M$, $Sym(M) \cong   \mathbb{C}[x_0,x_1]_{(x_0)}[y_0,y_1] $, which is clearly an graded ring over $\mathbb{C}[x_0,x_1]_{(x_0)}$. Therefore, we can consider the projective scheme $Proj(\mathbb{C}[x_0,x_1]_{(x_0)}[y_0,y_1])$.
$\implies

$ $Proj(\mathbb{C}[x_0,x_1]_{(x_0)}[y_0,y_1])$
$ \ \ \ \cong Spec\ \mathbb{C}[x_0,x_1]_{(x_0)}\ \times_{Spec\ \mathbb{C}} Proj(\mathbb{C}[ y_0,y_1])$
$ \\\\\\\\\\\\\\\  \ \   = Spec\ \mathbb{C}[x_0,x_1]_{(x_0)}\ \times Proj(\mathbb{C}[ y_0,y_1]) $
$ \\\\\\\\\\\\\\\  \ \   = Spec\ \mathbb{C}[x_0,x_1]_{(x_0)}\ \times \mathbb{P}^1$
So, we also have four affine open pieces of the scheme of $\mathbb{P} (\mathcal{E})$.
Now, we need to gluing such four affine pieces “compatible” with the structure in [Fulton,page8].(the meaning of “compatible” really confuses me ).
Recall that

$D_{+}(x_0) \bigcap D_{+}(x_1)$  $\cong  D_{+}(x_0x_1)$ $\cong  Spec \ \mathbb{C}[x_0,x_1]_{(x_0x_1)}$
$ Spec\ \mathbb{C}[x_0,x_1]_{(x_0)}\ \times_{Spec\ \mathbb{C}} D_{+}(y_0) $ $\cong$ $ Spec\ (\mathbb{C}[x_0,x_1]_{(x_0)} \bigotimes_{\mathbb{C}}\mathbb{C}[y_0,y_1]_{(y_0)})$ 

What shall I do ?  Should I need to construct ring isomorphisms between rings?
For example, an isomorphism $\mathbb{C}[x_0,x_1]_{(x_0x_1)}$  $\cong$  $\mathbb{C}[x_0,x_1]_{(x_0x_1)}$, obviously, we can find a trivial isomorphism, but is such isomorphism satisfies the “compatibility” requirement?
Thank you very much!!!

REPLY [10 votes]: The best  description of $\mathbb F_1$ is that it is just the blow-up of $\mathbb P^2$ at a point $P\in \mathbb P^2$, i.e $ \mathbb F_1=\tilde {\mathbb P^2}$.   
The exceptional curve $E\subset \tilde {\mathbb P^2}$ of the blow-up is the basis of the fibration $\pi:\mathbb F_1\to E=\mathbb P^1$ and the fibres $\tilde L$ of $\pi$ are the strict transforms of the lines $L\subset \mathbb P^2$  through $P$.
The exceptional curve $E$ corresponds to the quotient line bundle $\mathcal O$ in $\mathbb P(\mathcal O\oplus \mathcal O (1))$
If you make the blow-up explicit you will find  $\mathbb F_1$  embedded in $\mathbb P^2_{x:y:z}\times \mathbb P^1_{u:v}$ as the surface with equation $uy-vz=0 $ ( $\mathbb P^2_{x:y:z}$ has been blown up at $P=(1:0:0))$ .
The morphism $\pi_1$ sends the point of $((x:y:z), (u:v))\in \mathbb F_1$ to $(u:v)\in \mathbb P^1$ .