The Stacks project

Exercise 111.34.1. Consider the morphism of schemes

1. Show there does not exist a morphism $\sigma : S \to X$ such that $\pi \circ \sigma = \text{id}_ S$.

2. Show there does exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$.