geometric fibre    In algebraic geometry a geometric fiber is a
fiber of a bundle over a geometric point.  For a bundle p:EX of
topological spaces, the fibre over a point, xX may be thought of
as the preimage E x=p 1(x) equipped with its subspace topology.
More abstract, this is the pullback x× XE of p along the map
sending a singleton space * to xX, the object which is universal
with the property of making this diagram commute:  E x E p * x
X.\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 k¯ of k
and let ξ¯:Spec(k¯)X be a geometric point in X.   For a
morphism, p:EX, the geometric fibre over the geometric point x
is the pullback  E xE× XSpec(k¯).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 ‘