Fiber dimension theorem

For a dominant morphism of irreducible varieties, generic fiber dimension is source dimension minus target dimension, and this dimension occurs over a nonempty open subset. In particular, constant fiber dimension $d$ gives source dimension equal to target dimension plus $d$. For algebraic groups the equal-dimensional components and coset fibers allow the same dimension formula componentwise.

For an irreducible source, regard the morphism of algebraic varieties as dominant onto the closure of its image, with generic point $\eta$. The fiber dimension theorem gives the displayed formula. A proper morphism has closed image, so the closure can then be omitted.

Let $f:X\to Y$ be a proper morphism of irreducible algebraic varieties with equal dimension. If some fibre has an isolated point, its corresponding irreducible component has dimension zero. Apply the fiber dimension theorem to $X\to f(X)$: $0\ge \dim X-\dim f(X)$. Thus the closed image has dimension $\dim Y$ and equals $Y$. The incidence proof that a cubic surface contains a line is an application; the entire witness fibre need not be finite.

For a dominant morphism of algebraic varieties between irreducible varieties, there is a nonempty Zariski-open subset of the target over which every fibre is nonempty and every irreducible component has the displayed dimension. Apply Noether normalization to affine source charts over the target function field and clear denominators on the target to get a generic upper bound. Combine it with the closed-fibre lower bound in the fiber dimension theorem.

For a surjective morphism of irreducible varieties, choose an affine neighborhood of the target point and a finite Noether normalization to affine space of target dimension $r$. On an affine source neighborhood the composite fibre is cut out by $r$ functions, so repeated principal hypersurface dimension lemma gives dimension at least source dimension minus $r$ on every component. The normalization fibre is finite. The original fibre is therefore an open-and-closed union of components of that composite fibre, proving the bound without assuming target smoothness. For a reducible source the bound must instead be applied using the dimension of each source component.