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 2 under these conditions.