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
Full text: dvi.gz 18 k, dvi 39 k, ps.gz 390 k, pdf.gz 97 k, pdf 108 k.