(Fibre)
For \(f : X \to Y\) and \(y \in Y\) with residue field \(\kappa(y) = \OO_{Y,y}/\mfm_y\), the fibre is \[ X_y \da \fiberprod{X}{Y}{\Spec \kappa(y)} . \]
(Geometric points and fibres)
A geometric point of a scheme \(Y\) is a morphism \(\bar{y} \colon \Spec \Omega \to Y\) with \(\Omega\) an algebraically closed field. The geometric fibre of \(f \colon X \to Y\) over \(\bar{y}\) is \(X_{\bar{y}} = X \times_Y \Spec \Omega\). For a property \(P\) of schemes over a field such as connected, irreducible, reduced or integral, \(f\) has geometrically \(P\) fibres if every geometric fibre has \(P\), and a \(k\)-scheme \(X\) is geometrically \(P\) if \(X \times_k \Spec \bar{k}\) has \(P\).
\(X = \Spec \QQ(i) \to \Spec \QQ\) has a one-point fibre, which is integral, but its geometric fibre \(\Spec \QQ(i) \otimes_\QQ \overline{\QQ} \cong \Spec \overline{\QQ} \sqcup \Spec \overline{\QQ}\) is not connected. Over \(k = \FF_p(t)\), \(\Spec k[x]/(x^p - t)\) is integral, and after base change to \(\bar{k}\) it becomes \(\Spec \bar{k}[x]/((x - t^{1/p})^p)\), which is not reduced.
The fibre \(X_y\) is a scheme over \(\kappa(y)\); for \(f\) proper and \(\mcf\) coherent on \(X\), the groups \(H^i(X_y,\mcf|_{X_y})\) are finite-dimensional \(\kappa(y)\)-vector spaces. Base change to \(\Spec\OO_{Y,y}\) instead retains the part of the family over the generalisations of \(y\).
The underlying space of \(X_y\) is homeomorphic to \(f^{-1}(y)\), while the scheme structure records multiplicity and residue-field information. For \(\Spec \ZZ[i] \to \Spec \ZZ\), the fibre over \((p)\) is \(\Spec\FF_p[x]/(x^2+1)\): two points, one point with residue field \(\FF_{p^2}\), or one nonreduced point, according as \(p\) splits, is inert, or ramifies. Since \(\ZZ[i]\) is a free \(\ZZ\)-module of rank \(2\), each fibre has dimension \(2\) over \(\FF_p\), as does the generic fibre \(\Spec\QQ(i)\) over \(\QQ\); for a finite flat morphism \(f\) of Noetherian schemes, \(y\mapsto\dim_{\kappa(y)}\Gamma(X_y,\OO_{X_y})\) is locally constant on \(Y\).
Backlinks
T-MORFIBDIMFibre dimension and upper semicontinuityusesD-MORFLATFlat morphismsusesD-SCHFIBRESGeneric, special and central fibres; fat points; scheme-theoretic intersectionsusesPR-MORBCWhich properties are stable under base change and compositionuses