𝔸1-Connected Components of Affine Quadrics

Abstract For any smooth quadratic hypersurface X in A^n₊, we use the iterations of the functor of naive A^1-connected components S to study the field-valued sections of the sheaf of A^1-connected components ₀^A^{1} (X) of X. We prove that for any field F/k, the canonical isomorphism ₀^A^{1} (X) (F) ₍ S^n (X) (F) stabilizes at n=2, meaning that ₀^A^{1} (X) (F) =S^2 (X) (F). Furthermore, by combining this result with Morel’s characterization of A^1-connected spaces in terms of the triviality of field-valued sections of ₀^A^{1}, we provide a complete characterization of A^1-connected smooth quadratic hypersurfaces in A^n₊.