In algebraic geometry a geometric fiber is a fiber of a bundle over a geometric point.
For a bundle of topological spaces, the fibre over a point, may be thought of as the preimage equipped with its subspace topology. More abstract, this is the pullback of along the map sending a singleton space to , the object which is universal with the property of making this diagram commute:
Adapting this to the algebraic context we get the following definition.
Let be a scheme over some base field . Fix an algebraic closure of and let be a geometric point in .
For a morphism, , the geometric fibre over the geometric point is the pullback
Example and application
An important case is where is a finite étale cover, and then the geometric fibre is just a finite set.
Revised on September 25, 2012 13:26:25
by Urs Schreiber