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arXiv · 2312.05422

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

Neelarnab Raha
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Abstract

Let $X$ be a very general degree $d\geq 5$ 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 $d\geq3$ smooth hypersurface $X$ contains a line, we compute the nef cone of $X^{[n]}$ in a slice of the Néron-Severi space.

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Neelarnab Raha. 2023-12-09. Ample cones of Hilbert schemes of points on hypersurfaces in $\mathbb{P}^3$. https://arxiv.org/abs/2312.05422

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