  Let E be the restriction of the null-correlation bundle on $\mathbb{P}^{3}$
to a hyperplane. In this article, we show that the projective bundle
$\mathbb{P}(E)$ is isomorphic to a blow-up of a non-singular quadric in
$\mathbb{P}^{4}$ along a line. We also prove that for each $d \geq 2$, there
are hypersurfaces of degree d containing a line in $\mathbb{P}^{4}$ whose
blow-up along the line is isomorphic to the projective bundle over
$\mathbb{P}^{2}$.
