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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Automorphisms of Hilbert schemes of points on surfaces
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by Pieter Belmans, Georg Oberdieck and Jørgen Vold Rennemo PDF
Trans. Amer. Math. Soc. 373 (2020), 6139-6156 Request permission

Abstract:

We show that every automorphism of the Hilbert scheme of $n$ points on a weak Fano or general type surface is natural, i.e., induced by an automorphism of the surface, unless the surface is a product of curves and $n=2$. In the exceptional case there exists a unique nonnatural automorphism. More generally, we prove that any isomorphism between Hilbert schemes of points on smooth projective surfaces, where one of the surfaces is weak Fano or of general type and not equal to the product of curves, is natural. We also show that every automorphism of the Hilbert scheme of $2$ points on $\mathbb {P}^n$ is natural.
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Additional Information
  • Pieter Belmans
  • Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • MR Author ID: 1110715
  • Email: pbelmans@math.uni-bonn.de
  • Georg Oberdieck
  • Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Email: georgo@math.uni-bonn.de
  • Jørgen Vold Rennemo
  • Affiliation: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway
  • Email: jorgeren@uio.no
  • Received by editor(s): August 14, 2019
  • Received by editor(s) in revised form: December 10, 2019
  • Published electronically: June 24, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 6139-6156
  • MSC (2010): Primary 14C05, 14J50
  • DOI: https://doi.org/10.1090/tran/8106
  • MathSciNet review: 4155174