This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?



Minimal size of an open affine cover
Consider for simplicity smooth projective varieties defined over $\mathbb C$. In this case, the minimal size equals $n+1$ where $n$ is the dimension of the variety. 


Proof. Let $M^n$ be such a variety. Take $n+1$ generic very ample divisors $D\_0,...,D\_n$. Then such divisors don't have a common intersection. At the same time for any $i$ $M^n\setminus D\_i$ is affine. 
In order to show that you do need $n+1$ open affine sub-varieties, notice that the complement to an open affine sub-variety is a closed variety of dimension $n-1$. Now proceed by induction.


For projective irreducible varieties exactly the same reasoning should hold, and it can be generalised further I guess 


