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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: b.totaro@dpmms.cam.ac.uk
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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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