Take the 2-minute tour ×  I was doing some exercises in Liu's
book on Algebraic Geometry. I am currently trying to solve a
problem by showing the following:  Let $U \subset
\mathbb{P}^n_k$, k a field, be an affine open subset. Show that
the irreducible components of $\mathbb{P}^n_k-U$ all have
dimension n-1.  I would appreciate any help / hint here. I have
some problems understanding $\mathbb{P }^n_k$ at a deep (even
semi-deep) level. I suspect that one could maybe show that the
dimension of such an affine open should be of dimension n (or am
I wrong here?), since we can compute the dimension of X on any
open set. We should be able to write the complement as $V_+(I)$
for some homogenous ideal . However, I don't see how to get from
this to that the irreducible components of the component has
dimension n-1.  Thank you for looking at my question and please
forgive me if it's a naive one.  share|improve this question I
don't think it is naive. Here is something related:
mathoverflow.net/questions/22040/… –  Andre