Ample cones of Hilbert schemes of points on hypersurfaces in ${\mathbb{P}}^3$

Abstract

Let $X$ be a very general degree $d\ge 5$ hypersurface in ${\mathbb{P}}^3$. 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 $d\ge 3$ smooth hypersurface $X$ contains a line, we compute the nef cone of $X^{[n]}$ in a slice of the Néron-Severi space.

arXiv comment17 pages. Comments are welcome