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Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism $$(ev,\Phi):\...

... at least in the case where the target is a rationally connected variety. This question is a follow-up to question Constructing embedded families of curves with general moduli and Jason Starr's ...