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Pathologies on the Hilbert scheme of points

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We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Białynicki-Birula decomposition.

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I am very grateful to Piotr Achinger, Jarosław Buczyński and Maciek Zdanowicz for helpful and inspiring conversations and insightful comments of the early versions of this paper. The paper was prepared during the Simons Semester Varieties: Arithmetic and Transformations which is supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

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Correspondence to Joachim Jelisiejew.

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J. Jelisiejew Partially supported by NCN Grant 2017/26/D/ST1/00913.

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Jelisiejew, J. Pathologies on the Hilbert scheme of points. Invent. math. 220, 581–610 (2020).

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