Birational automorphisms of quartic Hessian surfaces
Authors:
Igor Dolgachev and JongHae Keum
Journal:
Trans. Amer. Math. Soc. 354 (2002), 30313057
MSC (2000):
Primary 14J28, 14J50; Secondary 11H56
Published electronically:
April 3, 2002
MathSciNet review:
1897389
Fulltext PDF Free Access
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]
Henry
F. Baker, Principles of geometry. Vol. III. Solid geometry:
Quadrics, cubic curves in space, cubic surfaces, Frederick Ungar
Publishing Co., New York, 1961. MR 0178392
(31 #2650a)
 [Bo]
Richard
Borcherds, Automorphism groups of Lorentzian lattices, J.
Algebra 111 (1987), no. 1, 133–153. MR 913200
(89b:20018), http://dx.doi.org/10.1016/00218693(87)902456
 [Cos]
François
R. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751. MR 716848
(85b:14049), http://dx.doi.org/10.1090/S00029947198307168484
 [CD]
François
R. Cossec and Igor
V. Dolgachev, Enriques surfaces. I, Progress in Mathematics,
vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969
(90h:14052)
 [Hu1]
J. Hutchinson, The Hessian of the cubic surface, Bull. Amer. Math. Soc., 5 (1889), 282292.
 [Hu2]
J. Hutchinson, The Hessian of the cubic surface, II, Bull. Amer. Math. Soc., 6 (1889), 328337.
 [Ke1]
Jong
Hae Keum, Every algebraic Kummer surface is the 𝐾3cover of
an Enriques surface, Nagoya Math. J. 118 (1990),
99–110. MR
1060704 (91f:14036)
 [Ke2]
Jong
Hae Keum, Automorphisms of Jacobian Kummer surfaces,
Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752
(98e:14039), http://dx.doi.org/10.1023/A:1000148907120
 [KK]
Jonghae
Keum and Shigeyuki
Kondō, The automorphism groups of Kummer
surfaces associated with the product of two elliptic curves, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1469–1487. MR 1806732
(2001k:14075), http://dx.doi.org/10.1090/S0002994700026313
 [Ko]
Shigeyuki
Kondō, The automorphism group of a generic Jacobian Kummer
surface, J. Algebraic Geom. 7 (1998), no. 3,
589–609. MR 1618132
(99i:14043)
 [Ni]
V.
V. Nikulin, Integer symmetric bilinear forms and some of their
geometric applications, Izv. Akad. Nauk SSSR Ser. Mat.
43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
(80j:10031)
 [PS]
I.
I. PjateckiĭŠapiro and I.
R. Šafarevič, Torelli’s theorem for algebraic
surfaces of type 𝐾3, Izv. Akad. Nauk SSSR Ser. Mat.
35 (1971), 530–572 (Russian). MR 0284440
(44 #1666)
 [Ro]
J. Rosenberg, Hessian quartic surfaces which are Kummer surfaces, math. AG/9903037.
 [Sa]
George
Salmon, A treatise on the analytic geometry of three dimensions.
Vol. II, Fifth edition. Edited by Reginald A. P. Rogers, Chelsea
Publishing Co., New York, 1965. MR 0200123
(34 #22)
 [To]
J.
A. Todd, A representation of the Mathieu group
𝑀₂₄ as a collineation group, Ann. Mat. Pura
Appl. (4) 71 (1966), 199–238. MR 0202854
(34 #2713)
 [vGe]
B. van Geemen, private notes 1999.
 [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), 133153. MR 89b:20018
 [Cos]
 F. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), 737751. 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), 282292.
 [Hu2]
 J. Hutchinson, The Hessian of the cubic surface, II, Bull. Amer. Math. Soc., 6 (1889), 328337.
 [Ke1]
 J.H. Keum, Every algebraic Kummer surface is the cover of an Enriques surface, Nagoya Math. J., 118 (1990), 99110. MR 91f:14036
 [Ke2]
 J.H. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math., 107 (1997), 269288. 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), 14691487. MR 2001k:14075
 [Ko]
 S. Kond The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geometry, 7 (1998), 589609. MR 99i:14043
 [Ni]
 V. Nikulin, Integral symmetric bilinear forms and some of their geometric applications, Math. USSR Izv. 14 (1980), 103167. MR 80j:10031
 [PS]
 I. PiatetskiShapiro, I.R. Shafarevich, A Torelli theorem for algebraic surfaces of type , Math. USSR Izv. 5 (1971), 547587. 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), 199238. MR 34:2713
 [vGe]
 B. van Geemen, private notes 1999.
Similar Articles
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, 20743 Cheongryangridong, Dongdaemungu, Seoul 130012, Korea
Email:
jhkeum@kias.re.kr
DOI:
http://dx.doi.org/10.1090/S0002994702030118
PII:
S 00029947(02)030118
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 KRF2000041D00014
Article copyright:
© Copyright 2002
American Mathematical Society
