Fiber of a morphism

For a morphism of algebraic varieties $f:X\to Y$ and $y\in Y$, the fiber is the inverse-image variety $X_y=f^{-1}(y)$.

For a dominant morphism of irreducible varieties, generic fiber dimension is source dimension minus target dimension, and this dimension occurs over a nonempty open subset. In particular, constant fiber dimension $d$ gives source dimension equal to target dimension plus $d$. For algebraic groups the equal-dimensional components and coset fibers allow the same dimension formula componentwise.

For a morphism of schemes $f:X\to Y$ and a point $y\in Y$, the scheme-theoretic fiber is the fiber product $X{\times }_Y\mathrm{Spec}\kappa (y)$. It remembers multiplicities and nilpotents that the underlying set-theoretic inverse image cannot detect.

Over a perfect field $k$ of characteristic $p$, the scheme-theoretic fiber of the Absolute Frobenius morphism of ${\mathbb{A}}_k^1$ over a rational point $a$ isIts underlying space is one point, but its coordinate ring has a nonzero nilpotent element, so the fibre is a length-$p$ nonreduced scheme.