The automorphism group of an affine quadric

• Autores: Burt Totaro
• Localización: Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, Vol. 143, Nº 1, 2007, págs. 1-8
• Idioma: inglés
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• Resumen

  • 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].