(Generic smoothness)
Let \(\characteristic k = 0\) and let \(f : X \to Y\) be a dominant morphism of varieties over \(k\) with \(X\) smooth. Then there is a nonempty open \(V \subseteq Y\) such that \(\ro{f}{f^{-1}(V)} : f^{-1}(V) \to V\) is smooth.
The theorem says that bad behaviour of a morphism is confined to a proper closed subset, so one may always shrink the base and assume smoothness. It is used to define ramification divisors, prove Sard-type statements, and reduce statements about a family to its general member.
Characteristic zero is essential: Frobenius gives a dominant morphism from a smooth variety that is smooth over no nonempty open. The proof in characteristic zero rests on separability of every field extension of \(k(Y)\), which is exactly what fails in characteristic \(p\).
The companion statement, generic flatness, has no characteristic-zero restriction: a finite type morphism to an integral Noetherian base is flat over a nonempty open.
Dependencies
D-MORSMSmooth morphisms and relative dimensionusesFE-MORFROBThe Frobenius morphism is nowhere smoothuses