Blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$ at a point.

Question

Let $x:= (0,0) \in \mathbb{P}^1 \times \mathbb{P}^1$. What is the blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$ at $x$?

Answer 1

This doesn't really have a special name, but the thing to notice is that blowing up takes the two lines through $x$ (one from each ruling) and separates them, so that the proper transforms $L_1$ and $L_2$ of these lines are $(-1)$-curves, and there is a third $(-1)$-curve linking them, namely the exceptional divisor $E$ (this follows from the general fact that blowing up a point on a curve in a surface decreases its self-intersection by $1$).

Now, Castelnuovo's contractibility criterion tells us that we can blow down any $(-1)$-curve to a smooth point. If we blow down $E$ we get back where we started, but if we blow down (without loss of generality) $L_2$, the proper transform of $E$ under the blowing down now has self intersection $0$. Meanwhile, the self-intersection of $L_1$ doesn't change so it is now the unique $(-1)$-curve. From here, it's not so hard to show that you have obtained the Hirzebruch surface $\mathbb F_1$, also known as the blow-up of $\mathbb P^2$ at a point.

Now you may know that $\mathbb P^1 \times \mathbb P^1$ is also known as $\mathbb F_0$, and the blow up/blow down procedure described above will (with a bit more care since $\mathbb F_{>0}$ are not totally homogeneous) in general transform $\mathbb F_e$ into $\mathbb F_{e+1}$. So while the blowup of $\mathbb P^1 \times \mathbb P^1$ at a point doesn't have a special name, it's the first in an infinite sequence of surfaces that form "roofs" relating all of the Hirzebruch surfaces (roof just refers to the shape of the relevant diagram).

Edited to add: some user has been trying to edit this answer today due to a misunderstanding. Please read more carefully; I do not claim that the blowup of $\mathbb P^1 \times \mathbb P^1$ is $\mathbb F_1$, but rather that one obtains the latter after also blowing down the proper transform of a line through the center of the initial blowup.