In this paper we study the following question: Is it true that a generic hypersurface X of degree d in P$^n+1$, where (d,n)≠ (3,1), (2,2), (3,2), does not admit a non-trivial, non-isomorphic surjective map to another smooth variety Y, except of course P$^n$? It is easy to see that it is true for n=1,2. We try to prove this for n=3 and can exclude all possibilities for Y except Y= G(1,4)∩ P$^6$and Y=V$_22^s$, a special Fano threefold of type V$_22$ found by Mukai and Umemura in [MU].