Let $ Y = V(xy-uv) \subset \mathbb{C}^4$ be a variety and consider the blowup $\pi: \tilde{X} \rightarrow Y$ at the point $(0,0,0,0)$. The exception set $Q$ is the projective quadric $V(xy-uv)\subset \mathbb{P}^3$ (Which we may identify as $\mathbb{P}^1\times \mathbb{P}^1$).
Consider the blow up of the ideals $(x,v)$ along $Y$: $\pi_X: X \rightarrow Y$. So set theoretically, we get $X = Cl(\{\big( (x,y,u,v), [x: v]\big): xy-uv =0 \})$. This also gives us a the blow-down $\tilde{X} \rightarrow X$.
The book that I am currently reading states the following: In $Q$, consider the curve $L$ given by $(y=v=0)$ . The image of $L$ in $X$ is then an isolated point.
How do we see this? My understanding is that the image of $L$ (which is in $Q$) in $X$ should intuitively by given by $(0,0,0,0) \times [x:0]$ a point. This point does not lie in the interior of $X$ and so must be isolated. But I have very little idea how to show this rigorously.
Any help given would be greatly appreciated!
Edit: The book I am referencing here is ‘Birational Geometry of Algebraic Varieties’ by Kollar-Mori. The example I am referring to can be found in pages 39 - 41, example 2.7