Nonstandard Generic Points

Starting front the Zariski topology, a natural notion of nonstandard generic point is introduced in complex algebraic geometry. The existence of this kind of point is a strong form of the Nullstellensatz. This notion is connected with the classical concept of generic point in the spectrum Spec(A(n,c)) of the corresponding algebra A(n,C). The nonstandard affine space *C(n) appears as all affine unfolding of the geometric space Spec(A(n,C)). This affine space is the disjoint union of the sets whose elements are the nonstandard generic points of prime and proper ideals of A(n,C): this structure leads to the definition of algebraic points in *C(n). A natural extension to analytic points in *C(n) is given by Robinson's concept of generic point in local complex analytic geometry. The end of this paper is devoted to a generalization of this point of view to the real analytic case.