The automorphism group of an affine quadric

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 {x0 + · · · + x 2 n = 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]. The background for these results is that almost all geometric tools work better for projective varieties than for affine varieties, because of the lack of compactness. Even basic questions like determining the automorphism group of an affine variety, or whether two affine varieties are isomorphic, can be difficult. Of course, some cases are easy. Consider an affine variety X −D where X is projective. If X is of general type, or more generally if the pair (X,D) is of log-general type in Iitaka’s sense (for example, when D is a smooth hypersurface of degree at least n + 2 in X = P), then the affine variety X −D has finite automorphism group [10, Theorem 11.12]. But when both X and D are of low degree in some sense, then the automorphism group of X −D is not at all understood. This justifies studying the basic case of affine quadrics. The key ingredient of the proof of the main Theorem 2.3 is that an anisotropic projective quadric with first Witt index equal to 1 is not ruled. More generally, a fundamental problem of birational geometry is to determine which varieties over a field are ruled. For example, Kollár proved that a large class of rationally connected complex hypersurfaces are non-rational by showing that they are not even ruled [16, Theorem V.5.14]. For anisotropic quadrics over a field, we give a conjectural answer to the problem of ruledness: they should be ruled if and only if the first Witt index is greater than 1 (Conjecture 3.1). Section 3 gives some evidence: in particular, the conjecture is true for quadratic forms of dimension at most 9 (thus for projective quadrics of dimension at most 7).