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

Abstract

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 $\mathbb{P}^{2}$.