Good Morning,
I'm trying to prove that two different definitions of the Hirzebruch Surfaces coincide, and am having problems. Let $a \geq 0$. My first definition for the $a^{th}$ surface is
$X_a= \mathbb{P}(\mathcal{O}(a) \oplus \mathcal{O}) \longrightarrow \mathbb{P}^1_{\mathbb{C}}$
My second definition is as follows. Let $C_a$ be a degree $a$ rational normal curve, ie the image under $\mathcal{O}(a)$ of $\mathbb{P}^1_{\mathbb{C}}$ into $\mathbb{P}^a_{\mathbb{C}}$, and let $D_a$ be the projective cone over $C_a$. That is, $D_a$ is defined by the same equations which define $C_a$, except now $D_a\subseteq \mathbb{P}^{a+1}$. Then $D_a$ is a surface which is smooth except for the possibly singular point $v=[0,…,0,1]$. Define $Y_a$ to be the blow-up of $D_a$ at $v$.
Why are $X_a$ and $Y_a$ isomorphic? (I want to stick to the algebraic or complex category, no smoothness allowed!)
Robert
Best Answer
The cone $D_a$ has a singular point of type $\frac{1}{a}(1,1)$ at its vertex. Blowing up the vertex, the exceptional divisor is a curve $C \subset Y_a$ isomorphic to $C_a$ and whose self-intersection is $\deg \mathcal{O}_{C_a}(-1)=-a$.
Since $Y_a$ is clearly a geometrically ruled surface over a rational curve (the ruling is given by the strict transform af the system of lines of $D_a$) and $C$ is a section of self-intersection $-a$, it follows $Y_a \cong X_a.$
Conversely, starting from the surface $X_a$ one can consider the unique section $C$ of negative self-intersection, namely $C^2=-a$; then this section can be blown down by Artin contractibility criterion.
The blow-down of $C$ is precisely the map $\varphi$ associated to the complete linear system $|C+aF|$ in $X_a.$
Indeed $h^0(X_a, C+aF)=a+2,$ hence $$\varphi \colon X_a \longrightarrow D_a \subset \mathbb{P}^{a+1}.$$
It is immediate to check that $\varphi$ is birational onto its image $D_a$, that it contracts $C$, that the ruling of $X_a$ is sent into a family of lines passing through the point $\varphi(C)$ and that a general hyperplane section of $D_a$ is a rational normal curve $C_a$ of degree $a$.
Therefore $D_a$ is a cone over $C_a$ and $X_a$ is isomorphic to the blow-up of $D_a$ at its vertex $\varphi(C)$.