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

DOI:
https://doi.org/10.1090/S0002-9947-02-03011-8

Published electronically:
April 3, 2002

MathSciNet review:
1897389

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 of rank 26 and signature . 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.

**[Ba]**H. Baker,*Principles of Geometry. Vol III*, Cambridge University Press, 1922, 2nd ed. 1954. MR**31:2650a****[Bo]**R. Borcherds,*Automorphism groups of Lorentzian lattices*, J. Algebra 111 (1987), 133-153. MR**89b:20018****[Cos]**F. Cossec,*Reye congruences*, Trans. Amer. Math. Soc. 280 (1983), 737-751. MR**85b:14049****[CD]**F. Cossec, I. Dolgachev,*Enriques surfaces*, Birkhaüser, Boston 1989. MR**90h:14052****[Hu1]**J. Hutchinson,*The Hessian of the cubic surface*, Bull. Amer. Math. Soc., 5 (1889), 282-292.**[Hu2]**J. Hutchinson,*The Hessian of the cubic surface, II*, Bull. Amer. Math. Soc., 6 (1889), 328-337.**[Ke1]**J.H. Keum,*Every algebraic Kummer surface is the**-cover of an Enriques surface*, Nagoya Math. J., 118 (1990), 99-110. MR**91f:14036****[Ke2]**J.H. Keum,*Automorphisms of Jacobian Kummer surfaces*, Compositio Math., 107 (1997), 269-288. MR**98e:14039****[KK]**J.H. Keum, S. Kond*The automorphism groups of Kummer surfaces associated with the product of two elliptic curves*, Trans. Amer. Math. Soc., 353 (2001), 1469-1487. MR**2001k:14075****[Ko]**S. Kond*The automorphism group of a generic Jacobian Kummer surface*, J. Algebraic Geometry, 7 (1998), 589-609. MR**99i:14043****[Ni]**V. Nikulin,*Integral symmetric bilinear forms and some of their geometric applications*, Math. USSR Izv. 14 (1980), 103-167. MR**80j:10031****[PS]**I. Piatetski-Shapiro, I.R. Shafarevich,*A Torelli theorem for algebraic surfaces of type*, Math. USSR Izv. 5 (1971), 547-587. MR**44:1666****[Ro]**J. Rosenberg,*Hessian quartic surfaces which are Kummer surfaces*, math. AG/9903037.**[Sa]**G. Salmon,*A Treatise on the Analytic Geometry of Three Dimensions, vol. 2*, W. Metcalfe and Son, Cambridge, 5th ed., Longmans and Green, London, 1915; reprint, Chelsea, New York, 1965. MR**34:22****[To]**J.A. Todd,*A representation of the Mathieu group**as a collineation group*, Ann. Mat. Pura Appl. (4) 71 (1966), 199-238. MR**34:2713****[vGe]**B. van Geemen, private notes 1999.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
14J28,
14J50,
11H56

Retrieve articles in all journals with MSC (2000): 14J28, 14J50, 11H56

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:
https://doi.org/10.1090/S0002-9947-02-03011-8

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