geometric point



The notion of geometric point refers to a certain kind of morphism between schemes in algebraic geometry; that is, to a specialized sort of generalized element of a scheme.

While a point in a topological space, XX, can be thought of as a continuous function from a singleton space to XX, in algebraic geometry the ‘spaces’ come with more structure as they are schemes and singletons correspond to the spectra of fields. Category-theoretically, one may think of any morphism into XX as a generalized point of XX, but when doing geometry it is often appropriate to restrict to a subclass of these to consider as the (less generalized) “points”.


Suppose a scheme SS is defined over a field kk, so is equipped with a morphism to Spec(k)Spec (k).


A geometric point ξ\xi in SS is a morphism from the spectrum Spec(k¯)Spec(\overline{k}) to SS where k¯\overline{k} is an algebraic closure/separable closure of kk.


In general the set of geometric points of a scheme is different from the set of ordinary points of its underlying topological space.


Revised on November 24, 2013 05:44:24 by Urs Schreiber (