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arXiv · 2304.12016

Brill-Noether theory of Hilbert schemes of points on surfaces

Arend Bayer·Huachen Chen·Qingyuan Jiang
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Abstract

We show that Brill--Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive, and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$ where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino's descriptionof the Hilbert--Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill--Noether loci given by nested Hilbert schemes.

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Arend Bayer, Huachen Chen, Qingyuan Jiang. 2023-04-24. Brill-Noether theory of Hilbert schemes of points on surfaces. https://arxiv.org/abs/2304.12016

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