Every fibre component has the expected lower bound

Given: The objects and hypotheses in the statement.

For a morphism $f:X\to Y$ of classical varieties and a closed point $y\in Y$, let $X_y=f^{-1}(y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_y=-\mathrm{\infty }$ if the fibre is empty. On affine charts $V\subseteq Y$ containing $y$ and $U\subseteq f^{-1}(V)$, writing $A=k[V]$ and $B=k[U]$, the fibre chart has coordinate ring $B\mathrm{/}\sqrt{{\mathfrak{m}}_{y}B}$. Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U\to V$. 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. (Reduced closed-point fibres and their dimension).

If $X$ is irreducible of dimension $n$ and $x\in X$ is a closed point, there are an affine neighborhood $U$ of $x$ and $n$ regular functions on $U$ whose common zero set is exactly $\{x\}$. 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. (A point is locally cut out by dim X functions).

Let $X$ be an irreducible classical variety of dimension $n$, and let $f_1,\dots ,f_r$ be global regular functions, with $r\ge 0$. Every nonempty irreducible component $Z$ of their common zero set has $\dim Z\ge n-r$. 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. (r equations lower dimension by at most r).

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).