Generic fiber of a scheme over a DVR

Question

What is (usually?) meant by the generic fiber of a scheme over a discrete valuation ring? I've seen in some talks now, could somebody give a precise definition?

Thank you very much in advance!

Answer 1

If $R$ is a discrete valuation ring, then $Y=\mathrm{Spec}(R)$ has two points, often denoted $\eta$ and $s$, the generic and the special (or closed) point, corresponding to the ideal $(0)$ and the unique maximal ideal $\mathfrak{m}_R$, respectively. The names are apt, as $\{\eta\}$ is dense in $Y$, while $\{s\}$ is closed in $Y$. Now a scheme over $Y$ is a scheme $X$ equipped with a morphism $f:X\to Y$. The generic (resp. special or closed) fibers of $X$ are the fibers over the generic (resp. closed) point of $Y$. As with any morphism of schemes, the fibers are equipped with scheme structures over the residue fields of the corresponding points, that is, $X_\eta$, the generic fiber, is a $k(\eta)=\mathrm{Frac}(R)$-scheme, while $X_s$, the special or closed fiber, is a scheme over the residue field $k(s)=R/\mathfrak{m}_R$.