Components of the nested Hilbert scheme of few points

Michele Graffeo; Paolo Lella

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arXiv · 2601.16765

Components of the nested Hilbert scheme of few points

Michele Graffeo·Paolo Lella

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Abstract

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth $n$-fold, for $n\geqslant 4$. Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of $\mathbb A^3$. Explicitly, we show that the Hilbert scheme of points on a hypersurface of $\mathbb{A}^3$ having a singularity of multiplicity at least 5 admits elementary components.

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Michele Graffeo, Paolo Lella. 2026-01-23. Components of the nested Hilbert scheme of few points. https://arxiv.org/abs/2601.16765

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