Projectivizations of Tautological bundles Let $Gr(k,n)$ be the Grassmannian of $k$-planes, and $Q$ its tautological quotient bundle. Is there a nice description of the projective bundle $P(Q)$ associated to $Q$? Is it a Grassmannian? Maybe if $k=2$?

Of course, it is not a Grassmannian (for instance, by the reason mentioned by Mohan), but it is a (partial) flag variety --- either $Fl(k,k+1;n)$ or $Fl(k,n-1;n)$, depending on which convention for $P(Q)$ is used.