A notion of seminormalization for real algebraic varieties

TL;DR

This paper introduces R-seminormalization, a new concept for real algebraic varieties, which adapts seminormalization to the real setting by focusing on real closed points and modifies singularities accordingly.

Contribution

The paper defines R-seminormalization for real algebraic varieties and explores its properties, extending previous complex results to the real case with several examples.

Findings

R-seminormalization is characterized by a universal property related to real closed points.

It modifies singularities by normalizing complex points and seminormalizing real points.

The approach adapts complex seminormalization techniques to real algebraic geometry.

Abstract

The seminormalization of an algebraic variety $X$ is the biggest variety linked to $X$ by a finite, birational and bijective morphism. In this paper we introduce a variant of the seminormalization, suited for real algebraic varieties, called the R-seminormalization. This object have a universal property of the same kind of the one of the seminormalization but related to the real closed points of the variety. In a previous paper, the author studied the seminormalization of complex algebraic varieties using rational functions that extend continuously to the closed points for the Euclidean topology. We adapt here some of those results to the R-seminormalization and we provide several examples. We also show that the R-seminormalization modifies the singularities of a real variety by normalizing the purely complex points and seminormalizing the real ones.