Generalized connectedness and Bertini-type theorems over real closed fields

TL;DR

This paper proves a real closed field analogue of Bertini's theorem, showing conditions under which hypersurface sections preserve formal reality and integrality in algebraic varieties over real closed fields.

Contribution

It introduces a new Bertini-type theorem over real closed fields, linking the sign of sections to the formal reality of their zero loci, extending classical results to real algebraic geometry.

Findings

Existence of open subsets of hypersurfaces preserving formal reality.

Characterization of sections with zero loci having formally real generic points.

Conditions under which sections do not change sign on the variety.

Abstract

In this paper, we establish a real closed analogue of Bertini's theorem. Let $R$ be a real closed field and $X$ a formally real integral algebraic variety over $R$. We show that if the zero locus of a nonzero global section $s$ of an invertible sheaf on $X$ has a formally real generic point, then $s$ does not change sign on $X$, and vice versa under certain conditions. As a consequence, we demonstrate that there exists a nonempty open subset of hypersurface sections preserving formal reality and integrality for quasi-projective varieties of dimension $\geq 2$ under these conditions.