Classifying the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$

Question

I'm working on an exercise where I'm supposed to note the different types of fibers on $V(uX^2 + vYZ) \subset \mathbb{P}^1 \times \mathbb{P}^2$. The way I was thinking to do this was to consider the following cases:

$X=0, Y = 0$

$X \neq 0, Y \neq 0$

one of $X=0$ or $Y=0$

and so on. I'm not really familiar with this kind of problem though. Is this the right approach?

On a side note, is this surface birational to something more familiar?

Answer 1

You should consider the projection $$ p:V=V(uX^2 + vYZ) \to \mathbb{P}^2$$ All fibers have one point, except those over $(0:1:0)$ and over $(0:0:1)$, which are copies of $\mathbb P^1$.
Hence $V$ is birational to $\mathbb{P}^2$.
More precisely $V$ is isomorphic to $\mathbb{P}^2$ blown up at these two exceptional points.