Generalized connectedness and Bertini-type theorems over real closed fields

TL;DR

This work extends Bertini-type results to real closed fields by linking the sign behavior of a global section $s$ of an invertible sheaf to the presence of a formally real generic point in its zero locus $V(s)$, under regularity and a curve-case conjecture. It develops a real-topological framework using the $X^{\mathrm{h}}$ topology and generalized connected subsets, along with convexity concepts, to control how signs and reality properties behave under deformations. The main contributions are (i) a real Bertini-type equivalence between sign consistency of $s$ and formal reality of $V(s)$, (ii) conditional interiority results for the set of sections with this property inside any finite-dimensional subspace $L$ of $\Gamma(X,\mathcal{L})$, and (iii) the existence of a nonempty open family of hypersurfaces that preserve formal reality and integrality for $\dim X \ge 2$, with unconditional results in the archimedean case. Together, these results provide Bertini-type tools in real algebraic geometry, enabling controlled real intersections and sign behavior of sections on varieties over real closed fields.

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.

Theorems & Definitions (20)

• Lemma 2.1: Van98, Chapter 1, Lemma 3.6
• Lemma 2.2: Van98, Chapter 7, §2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Conjecture 1
• Proposition 3.1