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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 $d \geq 2$, thereare hypersurfaces of degree d containing a line in $\mathbb{P}^{4}$ whoseblow-up along the line is isomorphic to the projective bundle over$\mathbb{P}^{2}$.

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