What does "generic fibre" mean in elliptic fibrations?

Question

I've just started to read about elliptic surfaces in algebraic geometry.

Here's a quote from wikipedia:

An elliptic surface is a surface that has an elliptic fibration [...] such that almost all fibers are smooth curves of genus 1. [...] This is equivalent to the generic fiber being a smooth curve of genus one.

I'm trying to understand what "generic fibre" means here.

I've found some other text which mentions some "fiber over the generic point", so I guess this must be the same thing (but I still don't know what it is).

So what's the precise definition of "generic fiber"? Are there any concrete, explicit examples?

Thanks!

(by the way, I have some background in algebraic varieties, but I'm still crawling my way into schemes)

Answer 1

Usually generic fiber refers to the fiber over the generic point. This also gives you examples. Just take for example the spectrum of a DVR $R$ and consider a morphism to $\text{Spec}(R)$ induced by the $\text{Spec}$ functor applied to your favorite morphism of rings $R \rightarrow A$. The other fiber in this setting is usually called the special fiber. You will find this situation (a morphism to the spectrum of a discrete valuation ring) while studying elliptic curves and elliptic fibrations etc when you are considering reductions.