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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

  1. $D_{+}(x_0) \bigcap D_{+}(x_1)$ $\cong D_{+}(x_0x_1)$ $\cong Spec \ \mathbb{C}[x_0,x_1]_{(x_0x_1)}$

  2. $ 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!!!

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1 Answer 1

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 $(u:v)\in \mathbb P^1$ .

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