Let $R$ denote a discrete valuation ring, so $Spec(R)$ consists of two points, the generic point and the special point.
Now I am familiar with the definition of fibers as a fibered product when considering a morphism $X \rightarrow SpecR$ for a scheme $X$ and a point $p$ in $SpecR$.
For the special fiber, we look at the special point of $SpecR$. But which morphism are we looking at? I've seen the term the special fiber lots of times. What is the special morphism corresponding to the special fiber?
I think I am missing something here..
Best Answer
Suppose $\pi: X \rightarrow \operatorname{Spec} R$ is a family over the spectrum of a DVR.
The ring $R$ has a unique maximal ideal $m$. Let $k=R/m$. Then there is a canonical quotient map $R \rightarrow k$ and dually a canonical morphism of schemes $\iota: \operatorname{Spec} k \rightarrow \operatorname{Spec} R$.
The special fibre of the family is then the fibre product $X \times_{\operatorname{Spec} R} \operatorname{Spec} {k}$.