Dimension is detected by avoiding linear subspaces

Given: The objects and hypotheses in the statement.

Let $X\subseteq {\mathbf{P}}_k^N$ be irreducible of dimension $d\ge 1$. If $f$ is homogeneous of positive degree and does not vanish identically on $X$, then $X\cap V_{+}(f)$ is nonempty and every irreducible component has dimension $d-1$. 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. (Nontrivial projective hypersurface sections).

Let $X,Y\subseteq {\mathbf{P}}_k^n$ be irreducible closed subvarieties. Every nonempty irreducible component $Z$ of $X\cap Y$ satisfies $\dim Z\ge \dim X+\dim Y-n$. If $\dim X+\dim Y\ge n$, then $X\cap Y\mathrm{\ne }\mathrm{\varnothing }$. 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. (Projective intersection dimension and nonemptiness).

If a Noetherian space $T$ is a finite union of closed subsets $T_1,\dots ,T_m$, then $\dim T={\max }_i\dim T_i$. For $m=0$ both sides are $-\mathrm{\infty }$. (Dimension of a finite closed union).

For every integer $n\ge 0$, $\dim {\mathbf{A}}_k^n=\dim {\mathbf{P}}_k^n=n$. 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. (Affine and projective n-space have dimension n).