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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?

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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?

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It is rather amazing that many people exist which know lots about blow-ups and are not aware of this description, btw :) –  Mariano Suárez-Alvarez May 13 '11 at 18:31
    
Well, I admit to not knowing much about blowups (they're shockingly rarely covered other than just blowup at a point) other than what I've learned "on the street"...though I do know that description, it isn't completely clear to me what the generalization is. Perhaps if I blow up along a $k$-plane, it's pairs $(q,\Lambda)$ with $\Lambda$ a $(k+1)$-plane containing the $k$-plane and $q$? –  Charles Siegel May 13 '11 at 18:56

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