Ample cones of Hilbert schemes of points on hypersurfaces in ℙ³

Let X X be a very general degree d ≥ 5 d≥ 5 hypersurface in P 3 P^3 . We compute the ample cone of the Hilbert scheme X [ n ] X[n] of n n points on X X for various small values of n n (the answer is already known for large n n ). We obtain complete answers in some cases and find lower bounds in certain others. We also observe that in the case of X [ 2 ] X[2] for quintic hypersurfaces X X , the existence (or absence) of hyperplane sections with points of high multiplicity also plays a role in the answer to the question at hand, in contrast with cases known earlier. Finally, in the case that a degree d ≥ 3 d≥ 3 smooth hypersurface X X contains a line, we compute the nef cone of X [ n ] X[n] in a slice of the Néron-Severi space.