Reduced closed-point fibres and their dimension

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 $B\mathrm{/}\sqrt{{\mathfrak{m}}_{y}B}$. Indeed the coordinate functions of $f{\mathrm{\mid }}_U:U\to V$ pull back ${\mathfrak{m}}_y$ to functions vanishing exactly at points over $y$; the relative radical-ideal/closed-set correspondence of Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals gives the reduced coordinate ring. General morphisms have the locally ringed-space meaning of Classical algebraic prevarieties, regular maps, and varieties, and the affine morphism test of A morphism from an open subset of a classical affine variety to an affine variety applies to $U\to V$.

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.