Smooth hypersurfaces of the blow-up.

Question

Let's assume $X$ is the blow up of $\mathbb{P}^n$ along a smooth subvariety $Z$. Especially $X$ is smooth. I was wondering what the hypersurfaces of $X$ look like? The hypersurfaces should give an ample divisor, so it should have positive intersection number with any curve. Hence I think this implies that it should contain the exceptional divisor which is a projective bundle over $Z$. So it seems like every hypersurface section has two components one should be the exceptional divisor and possibly another divisor (which is blow of some divisor in the projective space). But this seems to imply that hypersurfaces are not smooth (they have at least two components) and contradicts Bertini's theorem. (So a better question is what smooth hypersurfaces look like?)

Answer 1

I think you are asking a very difficult question. If you want to know the very ample divisors on $X$ then you are asking, at minimum, "what is the structure of the cone of ample/nef divisors on $X$?", and that is ignoring the ample/very ample distinction. If you Google the words "ample" and "blowup" together I think you will find some papers that will quickly demonstrate that it can take quite a bit of effort to say relatively little. Fixing $d > 1$, if you are blowing up a point $p$, then you are asking for something like a bound on the multiplicity of the divisor $D \subset \mathbb P^n$ at $p$ such that it can still be moved in a linear system $L \subset |D|$ which is large enough that strict transforms of members of $L$ cover $X$. And while this is straightforward enough for a single reduced point, blowing up at multiple and/or fat points will quickly get confusing, and blowing up a positive dimensional subvariety is even worse (in particular, the geometric characterization of a given linear system downstairs will become trickier and tricker). If you haven't already, I would advise that you start reading Lazarsfeld's Positivity in Algebraic Geometry to learn more about this kind of thing.