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?