For a very general hypersurface of degree d at least n+3 in P^{n+1}, any dominant rational map of prime degree to a smooth projective n-fold has a uniruled target, and the target is rationally connected when n is at most 3.
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Morphisms from a very general hypersurface
For a very general hypersurface of degree d at least n+3 in P^{n+1}, any dominant rational map of prime degree to a smooth projective n-fold has a uniruled target, and the target is rationally connected when n is at most 3.