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The ubiquity of smooth Hilbert schemes

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Abstract

We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials. These expressions naturally lead to an arrangement of nonempty Hilbert schemes as the vertices of an infinite full binary tree. Here we discover regularities in the geometry of Hilbert schemes. Specifically, under natural probability distributions on the tree, we prove that Hilbert schemes are irreducible and nonsingular with probability greater than 0.5.

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Acknowledgements

We especially thank Gregory G. Smith for his guidance in this research. We thank Mike Roth, Ivan Dimitrov, Tony Geramita, Chris Dionne, Ilia Smirnov, Nathan Grieve, Andrew Fiori, Simon Rose, and Alex Duncan for many discussions. We also thank the anonymous referee for helpful remarks that improved the paper. This research was supported by an E.G. Bauman Fellowship in 2011-12, by Ontario Graduate Scholarships in 2012–15, and by Gregory G. Smith’s NSERC Discovery Grant in 2015–16.

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  1. Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Andrew P. Staal

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  1. Andrew P. Staal
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Correspondence to Andrew P. Staal.

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Staal, A.P. The ubiquity of smooth Hilbert schemes. Math. Z. 296, 1593–1611 (2020). https://doi.org/10.1007/s00209-020-02479-8

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