geometric fibre

In algebraic geometry a *geometric fiber* is a fiber of a bundle over a geometric point.

For a bundle $p: E\to X$ of topological spaces, the fibre over a point, $x\in X$ may be thought of as the preimage $E_x = p^{-1}(x)$ equipped with its subspace topology. More abstract, this is the pullback ${x} \times_X E$ of $p$ along the map sending a singleton space $*$ to $x \in X$, the object which is universal with the property of making this diagram commute:

$\array{
E_x &\to& E
\\
\downarrow && \downarrow^{\mathrlap{p}}
\\
{*} &\stackrel{x}{\to}& X
}
\,.$

Adapting this to the algebraic context we get the following definition.

Let $X$ be a scheme over some base field $k$. Fix an algebraic closure $\overline{k}$ of $k$ and let $\overline{\xi} : Spec(\overline{k}) \to X$ be a geometric point in $X$.

For a morphism, $p: E\to X$, the **geometric fibre** over the geometric point $x$ is the pullback

$E_x \coloneqq E \times_X Spec (\overline{k})
\,.$

In EGA, the case is also considered in which the field is replaced by a local ring, (see EGA p. 112), in which case the word ‘geometric’ is dropped. (This needs checking.)

An important case is where $p$ is a finite étale cover, and then the geometric fibre is just a finite set.

- EGA I 3.4.5, p. 112.

Revised on September 25, 2012 13:26:25
by Urs Schreiber
(131.174.191.188)