For \(f: X \to Y\) and \(y \in Y\) with residue field \(\kappa(y) \da \OO_{Y,y}/\mfm_y\), the fibre is \[ X_y \da \fiberprod{X}{Y}{\Spec \kappa(y)} . \] Its underlying space is \(\inverseof{f}(y)\), but it carries a scheme structure over the field \(\kappa(y)\).
For \(\AA^1\slice k \to \AA^1\slice k\), \(t \mapsto t^2\), the fibre over a nonzero \(a\) is \(\Spec k[t]/(t^2 - a)\), which is two reduced points when \(a\) is a square in \(k\) and one point with residue field \(k(\sqrt a)\) otherwise. The fibre over \(0\) is \(\Spec k[t]/(t^2)\), one point of length \(2\). For \(\Spec \ZZ[i] \to \Spec \ZZ\) the fibre over \((p)\) is \(\Spec \FF_p[x]/(x^2+1)\): two points, one point, or a double point according to \(p \bmod 4\).
Fibres are a principal application of fibre products. The set-theoretic preimage loses the two things that make fibres behave: the multiplicity at a branch point, and the residue field extension at a point that splits only after base change. Both are visible in the examples, and in each the fibre has \(k\)-dimension \(2\) for every base point — this is the constancy that a naive count of preimages fails to see.
Flatness is the condition that controls this kind of fibrewise variation and preserves scheme-theoretic multiplicity under base change.
Dependencies
D-SCHFPRThe fibre product of schemesuses