# Pathologies on the Hilbert scheme of points

@article{Jelisiejew2019PathologiesOT, title={Pathologies on the Hilbert scheme of points}, author={Joachim Jelisiejew}, journal={Inventiones mathematicae}, year={2019}, volume={220}, pages={581-610} }

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.

## 21 Citations

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- 2019

The Bialynicki-Birula decomposition is generalized to singular schemes and an infinite family of small, elementary and generically smooth components of the Hilbert scheme of points of the affine four-space is found.

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