Real algebraic morphisms on 2-dimensional conic bundles

Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f : V → W can be approximated by regular maps in the space of 𝒞^∞ mappings from V to W in the 𝒞^∞ topology. Our main result is a complete solution to this problem in case W is the usual 2-dimensional sphere and V is a real algebraic surface of negative Kodaira dimension.