Projective fibre dimension is upper semicontinuous

Given: The objects and hypotheses in the statement.

A morphism $f:X\to Y$ of classical varieties is projective here if there is an integer $N\ge 0$ and a factorization $X\hookrightarrow Y\times {\mathbf{P}}_k^N\to Y$ in which the first map is a closed immersion and the second is projection. A closed immersion in this classical setting is an isomorphism onto a reduced closed subvariety. Morphisms have the general locally ringed-space meaning, checked on affine charts. 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 classical morphisms).

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).

For every classical variety $Y$ and $N\ge 0$, the projection $p:Y\times {\mathbf{P}}_k^N\to Y$ is a closed map. 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. (Projection from projective space over a variety is closed).

For a closed subset $Z\subseteq {\mathbf{P}}_k^N$ and integer $0\le r\le N$, $\dim Z<r$ if and only if some projective linear subspace of dimension $N-r$ is disjoint from $Z$. 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 is detected by avoiding linear subspaces).