Generalized connectedness and Bertini-type theorems over real closed fields

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 $\ge 2$ under these conditions.