Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T04:38:49.891Z Has data issue: false hasContentIssue false
  • English
  • Français

The automorphism group of an affine quadric

Published online by Cambridge University Press:  01 July 2007

BURT TOTARO
BURT TOTARO*
Affiliation:
DPMMS, Wilberforce Road, Cambridge CB3 0WB. e-mail: b.totaro@dpmms.cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Extract

We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko's in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x02+···+xn2=1} over the real numbers is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. This result is reminiscent of Wood's theorem that when n is a power of 2, every real polynomial mapping from the n-sphere to a lower-dimensional sphere is constant [22].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 1 - 8
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abhyankar, S.. On the valuations centered in a local domain. Amer. J. Math. 78 (1956), 321–348.CrossRefGoogle Scholar
[2]Ahmad, H. and Ohm, J.. Function fields of Pfister neighbors. J. Algebra 178 (1995), 653–664.CrossRefGoogle Scholar
[3]Baeza, R.. Quadratic Forms over Semilocal Rings. Lecture Notes in Math. vol. 655 (Springer, 1978).CrossRefGoogle Scholar
[4]Borel, A.. Linear Algebraic Groups (Springer, 1991).CrossRefGoogle Scholar
[5]Elman, R., Karpenko, N. and Merkurjev, A.. Algebraic and Geometric Theory of Quadratic Forms (American Mathematical Society, to appear).Google Scholar
[6]Gizatullin, M. and Danilov, V.. Automorphisms of affine surfaces, II. Math. USSR Izv. 11 (1977), 51–98.CrossRefGoogle Scholar
[7]Hoffmann, D.. Isotropy of quadratic forms over the function field of a quadric. Math. Z. 220 (1995), 461–476.CrossRefGoogle Scholar
[8]Hoffmann, D.. Splitting patterns and invariants of quadratic forms. Math. Nachr. 190 (1998), 149–168.CrossRefGoogle Scholar
[9]Hoffmann, D. and Laghribi, A.. Isotropy of quadratic forms over the function field of a quadric in characteristic 2. J. Alg. 295 (2006), 362–386.CrossRefGoogle Scholar
[10]Iitaka, S.. Algebraic Geometry (Springer, 1982).CrossRefGoogle Scholar
[11]Izhboldin, O.. Fields of u-invariant 9. Ann. Math. 154 (2001), 529–587.CrossRefGoogle Scholar
[12]Jelonek, Z.. Affine smooth varieties with finite group of automorphisms. Math. Z. 216 (1994), 575–591.CrossRefGoogle Scholar
[13]Karpenko, N.. On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15 (2000), 1–22.Google Scholar
[14]Knebusch, M.. Generic splitting of quadratic forms. I. Proc. London Math. Soc. 33 (1976), 65–93.CrossRefGoogle Scholar
[15]Knebusch, M.. Generic splitting of quadratic forms. II. Proc. London Math. Soc. 34 (1976), 1–31.Google Scholar
[16] J. Kollár. Rational Curves on Algebraic Varieties (Springer, 1999).Google Scholar
[17]Lam, T. Y.. Introduction to Quadratic Forms Over Fields (American Mathematical Society, 2005).Google Scholar
[18]Merkurjev, A.. Rost's degree formula. Notes of Lens mini-course, June 2001. http://www.math.ucla.edu/%7Emerkurev/publicat.htm.Google Scholar
[19]Ohm, J.. The Zariski problem for function fields of quadratic forms. Proc. Amer. Math. Soc. 124 (1996), 1679–1685.CrossRefGoogle Scholar
[20]Totaro, B.. Complexifications of nonnegatively curved manifolds. J. Eur. Math. Soc. 5 (2003), 69–94.CrossRefGoogle Scholar
[21]Totaro, B.. Birational geometry of quadrics in characteristic 2. J. Alg. Geom., to appear.Google Scholar
[22]Wood, R.. Polynomial maps from spheres to spheres. Invent. Math. 5 (1968), 163–168.CrossRefGoogle Scholar
[23]Yiu, P.. Quadratic forms between Euclidean spheres. Manuscripta Math. 83 (1994), 171–181.CrossRefGoogle Scholar