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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Birational automorphisms of quartic Hessian surfaces


Authors: Igor Dolgachev and JongHae Keum
Journal: Trans. Amer. Math. Soc. 354 (2002), 3031-3057
MSC (2000): Primary 14J28, 14J50; Secondary 11H56
Published electronically: April 3, 2002
MathSciNet review: 1897389
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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 Kondo. 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
Email: idolga@umich.edu

JongHae Keum
Affiliation: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea
Email: jhkeum@kias.re.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03011-8
PII: S 0002-9947(02)03011-8
Keywords: Quartic Hessian surfaces, automorphisms, K3 surface, Leech lattice
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
Article copyright: © Copyright 2002 American Mathematical Society