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 P³. 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.