Projective bundles and blowing ups

Abstract

We study the blowing up $\widetilde{X}$ of a smooth projective variety $X$ along a smooth center $B$ that is equipped with a projective bundle structure over a variety $Z$. If $B$ is a point, then $X$ is a projective space. If the Picard number $\rho (X)$ is $1,$ then $\dim Z$ has a lower bound $\dim X-\dim B-1.$ Moreover, when $\dim Z$ is $\dim X-\dim B-1,$ $X$ is a projective space and $B$ is a linear subspace in $X.$ If $X$ is a projective space $¶_n$ and $B$ is a curve, then either $n$ is $3$ and $B$ is a twisted cubic curve or $n$ is an arbitrary integer and $B$ is a line in $¶_n$. If $X$ is a quadric $Q_n$ and $B$ is a curve, then $n$ is $3$ and $B$ is a line in $Q_3$.