TITLE: Natural description of fibers of blowup of projective space along a subspace
QUESTION [5 upvotes]: Let $\mathbb{P}^n$ be a projective space, and let $\mathbb{P}^k$ a linear subspace.  There are many descriptions of $Bl_{\mathbb{P}^k}\mathbb{P}^n$, but I haven't seen one that's really intrinsic, they tend to rely on choosing a subspace complementary to the chosen linear one.
So is there any natural description of the fiber of the blowup over a point in the linear subspace? Something like the set of lines through that point satisfying a property, or the like? As mentioned, I'm interested mostly in the case of a linear subspace, but is there a way that generalizes to other subvarieties?

REPLY [3 votes]: When you blow up a single point $p$, the blow-up is the set of pairs $(q,\ell)$ with $q$ a point in $\mathbf P^n$ and $\ell$ a line joining $p$ and $q$ (and the map to the original space is just the projection onto the first coordinate). Can you see how to generalize this?