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A support theorem for nested Hilbert schemes of planar curves

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Abstract

Consider a family of integral complex locally planar curves. We show that under some assumptions on the base, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.

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Notes

  1. By having the same singularities, we mean that the completions of local rings at two corresponding singular points (see Definition 1.1) are isomorphic.

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Acknowledgements

I wish to thank my supervisor Luca Migliorini for suggesting me this problem as long as for the countless comments and corrections. Then I would like to thank Filippo Viviani and Gabriele Mondello for the helpful comments and suggestions. I also thank the anonymous reviewers for pointing out a mistake in the previous version of this paper as long as several imprecisions and misprints. Finally I am grateful to Enrico Fatighenti, Danilo Lewanski, Giovanni Mongardi and Marco Trozzo for the support in writing this article.

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  1. Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Camilla Felisetti

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  1. Camilla Felisetti
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Correspondence to Camilla Felisetti.

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Felisetti, C. A support theorem for nested Hilbert schemes of planar curves. manuscripta math. 164, 467–488 (2021). https://doi.org/10.1007/s00229-020-01189-z

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  • DOI: https://doi.org/10.1007/s00229-020-01189-z

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