My question concernes following former thread in MO: https://mathoverflow.net/questions/117808/the-intersection-multiplicity-of-the-canonical-divisor-of-a-surface-with-a-fibre
here we have a (minimal) surface S , a smooth curve C and a morphism $p:S \to C $ with generic fibre isomorphic to $P^1$. let futhermore $F$ be an arbitrary closed fibre. I want to know why the self intersection $F^2$ is zero. A user's argument was that as closed fibres are algebraically equivalent.
Firstly why does it hold and why algebraically equivalence implies the numerically one?