Nonstandard Generic Points

Abstract

Starting from 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 $\text{Spec}({\cal A}_{n,\mathbb C})$ of the corresponding algebra ${\cal A}_{n,\mathbb C}$. The nonstandard affine space ${^*\mathbb C}^n$ appears as an affine unfolding of the geometric space $\text{Spec}({\cal A}_{n,\mathbb C})$. This affine space is the disjoint union of the sets whose elements are the nonstandard generic points of prime and proper ideals of ${\cal A}_{n,\mathbb C}$: this structure leads to the definition of algebraic points in ${^*\mathbb C}^n$. A natural extension to analytic points in ${^*\mathbb C}^n$ 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.