A notion of seminormalization for real algebraic varieties

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 has a universal property of the same kind as 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 some of those results here 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.