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Galois closures and elementary components of Hilbert schemes of points

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Abstract

Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements.

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Notes

  1. Also referred to as the \(S_n\)-closure.

  2. We wholeheartedly thank the anonymous referee for noting that Theorem 1.1 should hold in greater generality.

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Acknowledgements

It is a pleasure to thank Joachim Jelisiejew for comments on an earlier draft of this paper. We are deeply indebted to the anonymous referee whose suggestions both shortened our arguments and generalized our results. The referee greatly simplified our proofs of Propositions 4.6 and 4.9, and noted that our results should extend to the case when \(m\ne n-1\).

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  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

    Matthew Satriano & Andrew P. Staal

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  1. Matthew Satriano
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  2. Andrew P. Staal
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Correspondence to Matthew Satriano.

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MS was partially supported by a Discovery Grant from the National Science and Engineering Research Council of Canada and a Mathematics Faculty Research Chair.

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Satriano, M., Staal, A.P. Galois closures and elementary components of Hilbert schemes of points. Sel. Math. New Ser. 30, 28 (2024). https://doi.org/10.1007/s00029-024-00915-9

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