Correspondence between projective bundles over P2 and rational hypersurfaces in P4

Abstract

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