Example sought: disconnected closures of fibers of line bundles

Question

I'm looking for a simple example of the following (so that I can get better intuition for it). If possible, I'd like a 2 dimensional rational example.

$X$ is an irreducible projective variety, with an open subset $U$ that is (isomorphic to) the total space of a line bundle over an irreducible variety - in other words such that there exists an irreducible $Y$ where $\pi: U \rightarrow Y$ is a line bundle. We can look at $Z=\bigcup_{y\in Y} (\overline{\pi^{-1}(y)}\setminus \pi^{-1}(y))$ (where the closure is taken in $X$) - the union of the ''limits'' of the fibers of $\pi$. I want an example where $Z$ is not connected.

The intuition is that the ''limit'' of a fiber over a special point $y$ in $Y$ might actually be ''far away'' from the ''limits'' of the fibers of all the points ''close to'' $y$.

Answer 1

Take the blowup $X = \text{Bl}_p(\mathbb{P}^1\times \mathbb{P}^1)$ with exceptional $E$ and $l,l' \subset X$ the proper transforms of the rulings through $p \in \mathbb{P}^1\times \mathbb{P}^1$. Then $U := X\setminus (E\cup l)$ is the trivial line bundle over $\mathbb{P}^1$ but $Z$ will be the disconnected set $l\cup (l'\cap E) \setminus (l\cap E)$.