Abstract
This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general fibers are rational curves. Assume that the set of real points of $X$ is an orientable 3-manifold $M$. The aim of the paper is to give a topological description of $M$. It is shown that $M$ is either Seifert fibered or a connected sum of lens spaces. Much stronger results hold if $S$ is rational.
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