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?


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


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.


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