AMPLE CONES OF HILBERT SCHEMES OF POINTS ON HYPERSURFACES IN P3

Let X be a very general degree d >= 5 hypersurface in P-3 . We compute the ample cone of the Hilbert scheme X ([n]) of n points on X for various small values of n (the answer is already known for large 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] for quintic hypersurfaces 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 smooth hypersurface X contains a line, we compute the nef cone of X ([n]) in a slice of the Neron-Severi space.