Fibres have pure expected dimension over a dense open

Given: The objects and hypotheses in the statement.

For a dominant morphism $f:X\to Y$ between irreducible classical varieties and every closed point $y\in Y$, each nonempty irreducible component $Z$ of $X_y$ satisfies $\dim Z\ge \dim X-\dim Y$. No bound is asserted for an empty fibre. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Every fibre component has the expected lower bound).

Let $f:X\to Y$ be dominant between irreducible affine varieties, put $A=k[Y]\subseteq B=k[X]$, and let $r={\mathrm{trdeg}}_{k(Y)}k(X)$. There are $0\mathrm{\ne }a\in A$ and elements $t_1,\dots ,t_r\in B_a$, algebraically independent over $A_a$, such that $B_a$ is module-finite over $A_a[t_1,\dots ,t_r]$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (A dominant affine map factors finitely over relative affine space after shrinking the base).

The image of a dominant morphism $f:X\to Y$ between irreducible affine varieties contains a nonempty principal open subset of $Y$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dominant affine images contain a principal open).

Every classical variety is Noetherian and has finitely many irreducible components. Every open or closed subvariety has a finite affine cover. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Classical varieties have finite irreducible decompositions).

For every open cover $T={\bigcup }_{i\in I}{U}_i$ of a Noetherian space, $\dim T={\sup }_i\dim U_i$, with empty supremum $-\mathrm{\infty }$. (Dimension can be computed on an open cover).

Let $k\subseteq K\subseteq L$ be a tower of field extensions. Assume that ${\mathrm{trdeg}}_{k}K$ and ${\mathrm{trdeg}}_{K}L$ are finite. Then ${\mathrm{trdeg}}_{k}L={\mathrm{trdeg}}_{k}K+{\mathrm{trdeg}}_{K}L\mathrm{.}$ (Transcendence degree is additive in finite towers).

Let $k$ be a field, let $A$ be a finite-type $k$-domain, and let $K=\mathrm{Frac}(A)$. Then $\dim A={\mathrm{trdeg}}_{k}K\mathrm{.}$ (Affine-domain dimension equals transcendence degree).

If $U$ is a nonempty open of an irreducible classical variety $X$, then $\dim U=\dim X$. Every proper closed subvariety $Z⊊X$ has $\dim Z<\dim X$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

If $X$ is an irreducible classical variety, then $\dim X={\mathrm{trdeg}}_{k}k(X)<\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Dimension equals transcendence degree).

For irreducible classical $X$, the fraction fields of all nonempty affine charts identify canonically; denote the resulting field by $k(X)$. A dominant morphism $f:X\to Y$ between irreducible classical varieties induces an injection $f^{\ast }:k(Y)\hookrightarrow k(X)$. Dominant means that the image is dense. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Function fields and dominant pullbacks on general varieties).