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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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Birational automorphisms of quartic Hessian surfaces
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by Igor Dolgachev and JongHae Keum PDF
Trans. Amer. Math. Soc. 354 (2002), 3031-3057 Request permission

Abstract:

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice $II_{1,25}$ of rank 26 and signature $(1,25)$. The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondō. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.
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Additional Information
  • Igor Dolgachev
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 58860
  • Email: idolga@umich.edu
  • JongHae Keum
  • Affiliation: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea
  • MR Author ID: 291447
  • Email: jhkeum@kias.re.kr
  • Received by editor(s): June 30, 2001
  • Received by editor(s) in revised form: August 27, 2001
  • Published electronically: April 3, 2002
  • Additional Notes: Research of the first author was partially supported by NSF grant DMS 9970460.
    Research of the second author was supported by Korea Research Foundation Grant KRF-2000-041-D00014
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 3031-3057
  • MSC (2000): Primary 14J28, 14J50; Secondary 11H56
  • DOI: https://doi.org/10.1090/S0002-9947-02-03011-8
  • MathSciNet review: 1897389