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