Affine intersection bound via the diagonal

Given: The objects and hypotheses in the statement.

Products of nonempty classical varieties exist in the category of classical varieties, and $\dim (X{\times }_{k}Y)=\dim X+\dim Y$. If both factors are irreducible, their product is irreducible. 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. (Dimensions add under products).

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 $X\subseteq {\mathbf{A}}^n$ is an affine variety, then the diagonal in $X\times X$ is cut out by $x_i\otimes 1-1\otimes x_i$ for $1\le i\le n$. (The affine diagonal is cut out by coordinate differences).