Global and local dimension of classical varieties

For a classical variety $X$, let $\dim X$ be its chain dimension. If $X_1,\dots ,X_m$ are its irreducible components and $x\in X$ is a closed point, define ${\dim }_{x}X={\max }_{x\in X_i}\dim X_i$. The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $-\mathrm{\infty }$.

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.