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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: [email protected]
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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

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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