Hilbert specialization of parametrized varieties

TL;DR

This paper extends Hilbert specialization from polynomials to prime ideals and affine varieties, and explores specialization at polynomial values, with applications to irreducibility in algebraic geometry.

Contribution

It introduces a novel extension of Hilbert specialization to prime ideals and affine varieties, and studies specialization at polynomial values.

Findings

Extended Hilbert specialization to prime ideals and affine varieties.

Applied specialization to analyze irreducibility of variety intersections.

Explored specialization at polynomial values beyond scalar cases.

Abstract

Hilbert specialization is an important tool in Field Arithmetic and Arithmetic Geometry, which has usually been intended for polynomials, hence hypersurfaces, and at scalar values. In this article, first, we extend this tool to prime ideals, hence affine varieties, and offer an application to the study of the irreducibility of the intersection of varieties. Then, encouraged by recent results, we consider the more general situation in which the specialization is done at polynomial values, instead of scalar values.