Birationality and isomorphism of Hirzebruch surfaces Question 1: How does one show that the $n$-th Hirzebruch surface 
$$\mathcal{H}_n:=V(u^nX-v^nY)\subset \mathbb{P}^2\times \mathbb{P}^1$$
($[u:v]\in \mathbb{P}^1, [X:Y:Z]\in \mathbb{P}^2$)
and the projective product $\mathbb{P}^1\times \mathbb{P}^1$ are birational? I am having trouble finding which open set of $\mathcal{H}_n$ to make isomorphic to $\mathbb{A}^1\times \mathbb{A}^1$. 
Question 2: Also, I want to understand how $\mathcal{H}_n$ is isomorphic to the blow-up of $\mathbb{P}^2$ at some point $p$. I understand that the latter is a geometrically ruled surface and should be isomorphic as such, but can one construct an explicit isomorphism (regular map with inverse) between the two?
Question 3: In a distinguished open set, $\mathcal{H}_n$ is also a line bundle over the projective line. How does one show $\mathcal{H}_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(n)\bigoplus \mathcal{O}_{\mathbb{P}^1})$?
Are there any useful references involving Hirzebruch surfaces that I can read to learn more? 
 A: Question 2. Let $Y=\textrm{Bl}_P\mathbb P^2$, the blow-up at a point $P\in\mathbb P^2$. To convince you that $Y$ is a projective bundle of rank 1 over $\mathbb P^1$, you can project $Y$ onto the exceptional divisor $E\cong\mathbb P^1$ and see that the fibers are $\mathbb P^1$'s, because they are the strict transforms of lines through $P$. This is not quite what you want, though - it was just to make a remark a posteriori.
Let $\mathscr E=\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(1)$ and $X=\mathbb P(\mathscr E)$. We want to show that $X\cong Y$. We have
$$Y=\{((x_0,x_1,x_2);(u,v))\in \mathbb P^2\times\mathbb P^1\,|\,x_1v=x_2u\}\subset \mathbb P^2\times\mathbb P^1.$$ Let us take the standard open covering $(U_0,U_1)$ of $\mathbb P^1$; if $p:\mathscr E\to \mathbb P^1$ comes with trivializations $\phi_i:p^{-1}(U_i)\to U_i\times \mathbb A^2$, and hence with transition maps $\phi_{ij}:U_{ij}\times\mathbb A^2\to U_{ij}\times\mathbb A^2$, then on $\mathbb P(\mathscr E)\to \mathbb P^1$ we have the same (but projectivized) transition functions $\phi_{ij}:U_{ij}\times\mathbb P^1\to U_{ij}\times\mathbb P^1$. In our case, we only have two open sets, so there is only one isomorphism which we have to use in order to get $X$ by gluing two copies of $U_0\times\mathbb P^1$ and $U_1\times \mathbb P^1$ together: for instance, by using the isomorphism $$\phi_{01}:U_{01}\times\mathbb P^1\to U_{01}\times\mathbb P^1$$ sending $(z,(u,v))\mapsto (1/z,(u,zv))$, we can construct open immersions
\begin{align}
U_0\times\mathbb P^1\to Y\,\,\,\,\,\,\,\,\,\,(z,(u,v))\mapsto ((u,zv,v),(z,1)) \notag\\
U_1\times\mathbb P^1\to Y\,\,\,\,\,\,\,\,\,\,(z,(u,v))\mapsto ((u,v,zv),(1,z)) \notag\\
\end{align}
which glue to an isomorphism $\mathbb P(\mathscr E)\to Y$, by construction.
