What is a fiber over a point?

Question

Let $X\to S$ be a morphism of schemes.

First what does it mean to be a fiber over a point? Is it like the presage of $s\in S$ under the morphism?

Sometimes I see $X_s$ and sometimes I see $X_\bar{s}$. What are the differences?

Answer 1

Hartshorne defines the fibre of the morphism $f: X \to S$ over the point $s \in S$ to be the scheme $$ X_s = X \times_S \text{Spec} k(s).$$

He observes that this is a scheme over $k(s)$, and provides an exercise (3.10) in which one can show that its underlying topological space is homeomorphic to the subset $f^{-1}(s) \subset X$.

If you are unsure of any of this, I recommend you review Chapter 2 Section 3 of Hartshorne, or the equivalent section of Vakil's notes/Gortz-Wedhorn.

I am unsure what you mean by $X_{\overline{s}}$.