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 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 $II_{1,25}$ 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.
- 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
- Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI https://doi.org/10.1016/0021-8693%2887%2990245-6
- François R. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751. MR 716848, DOI https://doi.org/10.1090/S0002-9947-1983-0716848-4
- François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969
- J. Hutchinson, The Hessian of the cubic surface, Bull. Amer. Math. Soc., 5 (1889), 282-292.
- J. Hutchinson, The Hessian of the cubic surface, II, Bull. Amer. Math. Soc., 6 (1889), 328-337.
- Jong Hae Keum, Every algebraic Kummer surface is the $K3$-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99–110. MR 1060704, DOI https://doi.org/10.1017/S0027763000003019
- Jong Hae Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752, DOI https://doi.org/10.1023/A%3A1000148907120
- Jonghae Keum and Shigeyuki Kond\B{o}, 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, DOI https://doi.org/10.1090/S0002-9947-00-02631-3
- Shigeyuki Kond\B{o}, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geom. 7 (1998), no. 3, 589–609. MR 1618132
- 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
- I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type ${\rm K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
- J. Rosenberg, Hessian quartic surfaces which are Kummer surfaces, math. AG/9903037.
- George Salmon, A treatise on the analytic geometry of three dimensions. Vol. II, 5th ed., Chelsea Publishing Co., New York, 1965. Edited by Reginald A. P. Rogers. MR 0200123
- J. A. Todd, A representation of the Mathieu group $M_{24}$ as a collineation group, Ann. Mat. Pura Appl. (4) 71 (1966), 199–238. MR 202854, DOI https://doi.org/10.1007/BF02413742
- 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
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
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