Assume that $ f : X \to Y $ is a morphism of schemes. Under what situations can we find some non empty open affine subscheme $ {\rm Spec} B \subseteq Y$ such that $ f^{-1}({\rm Spec}B)$ is affine? Here is what I have already thought about:

- If $ f $ is an affine morphism, we just take any open affine subscheme of $Y$.
- Let $X$ be the affine line with two origins. If we think about the morphism $ X \to \mathbb{A}^1$, then there is no affine neighborhood of the origin in $ \mathbb{A}^1$ whose preimage is affine. However, the preimage of $ \mathbb{A}^1 \backslash \{ 0 \} $ is affine.
- We can't find such a $ {\rm Spec} B $ for the morphism $ \mathbb{P}^1_k \to {\rm Spec} k $.

nontrivialexamples and characterizations? – Martin Brandenburg Sep 22 '13 at 21:34