Correspondence between Projective bundles over ${\mathbb{P}}^2$ and rational Hypersurfaces in ${\mathbb{P}}^4$

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 $d\ge 2$, 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 ${\mathbb{P}}^2$.