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.