double cover over $\mathbb{P}^2$ ramified on conics I am reading the book Compact Complex Surfaces. At Section V.22, it states that the double covering $X\rightarrow\mathbb{P}^2$ branched over a smooth conic is a quadric surface.
I am able to compute the Hodge numbers: $h^{0,0}(S)=h^{2,2}(S)=1,h^{1,1}(S)=2$ and others are zero.
By the classification of surfaces, it should be a Hirzebruch surface $\mathbb{F}_n$ for some $n\geq0$.
How can I identify the $n$?

Just note that if $f(x_1,x_2,x_3)$ is the equation of a smooth conic, then the double covering is given by the equation
$$
x_0^2 = f(x_1,x_2,x_3),
$$
where $x_0$ is an extra variable. Obviously, this is a smooth quadric.
