Burt Totaro: The automorphism group of an affine quadric


Submission: 2006, Jul 24

We determine the automorphism group for a large class of affine quadrics, viewed as affine algebraic varieties. The proof uses Karpenko's theorem that an anisotropic quadric with first Witt index equal to 1 is not ruled, as well as a general result on automorphisms of affine varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere 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. We also conjecture a converse to Karpenko's theorem, predicting exactly which quadrics are ruled.

2000 Mathematics Subject Classification: Primary 11E04, Secondary 14R20.

Keywords and Phrases: Quadratic forms, affine algebraic geometry, ruled varieties

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