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



Hirzebruch Surfaces
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 $\varphi(C)$. 


