Liu's book define regular fibered surface X$\to$S is relative minimal surface if it does not contain any exceptional divisor, regular fibered surface X $\to$ S is minimal surface if every birational map of regular fibered S-surfaces Y $\to$ X is a birational morphism. It is hard for me to understand them, here are my questions:
Given a regular fibered surface, how to judge whether it is relative minimal or minimal?
Does any typical examples to show what they look like?
Why we call it minimal here ? Does any relation with the minimal surface in the differential geometry?
Let $X$ be a smooth compact complex surface.
ad 1: To check whether X has no exceptional divisors one can often apply the adjunction formula. Examples: a) A K3-surface $X$ has trivial canonical divisor, the adjunction formula gives $2g(C) - 2 = C^2$ for any curve $C$ on $X$, hence no -1 curves exist. b) $X = \mathbb P^2$. Each curve $C$ on $\mathbb P^2 $ has positive self intersection number; hence one does not even need the adjunction formula to exclude exceptional divisors.
To determine whether an exceptional divisor $C$ of a surface $X$ is contained in a fibre $F$ of a regular fibration $X \longrightarrow S$ one checks whether $(C,F) = 0$ for the intersection number.
ad 2: A typical example of a surface $X$ with an exceptional divisor is the blow-up of $\mathbb P^2$ in one point; one gets the first Hirzebruch surface $X = \Sigma _1$. Examples of regular fibrations $X \longrightarrow S$ with fibres having exceptional curves are fibrations, which result from a pencil on $X$ with base points and the corresponding rational map $X \longrightarrow \mathbb P^1$, which is undefined at the base points. Blowing up the base points may produce a fibration with exceptional fibres. For an example see About fibers of an elliptic fibration.
ad 3: Minimal in complex analysis and algebraic geometry refers to blowing-up and its inverse blowing-down. These methods are very important because they conserve smoothness. Blowing-down is considered a kind of simplifying, e.g., it reduces the Picard group. A surface is minimal when no further blowing-down is possible. Every birational map between surfaces $f:X \longrightarrow Y $ factors as a finite sequence of blow-downs. If $X$ has no exceptional divisors then $f$ is already a morphism.
I do not know any relation to the concept of minimality in differential geometry.
For the domain of algebraic geometry the concept is introduced and explained in the textbook "Hartshorne, R.: Algebraic geometry." See also the keywords "monoidal transformation" and "$\sigma$-process" in his book.