For any smooth quadratic hypersurface $X$ in $\mathbb A^n_k$, we use the iterations of the functor of naive $\mathbb{ A}^1$-connected components $\mathcal{ S}$ to study the field-valued sections of the sheaf of $\mathbb{ A}^1$-connected components $\pi_0^{ \mathbb{ A}^1}(X)$ of $X$. We prove that for any field $F/k$, the canonical isomorphism $\pi_0^{ \mathbb{ A}^1}(X)(F) \xrightarrow{ \sim} \lim_{ n} \mathcal{ S}^n(X)(F)$ stabilizes at $n=2$, meaning that $\pi_0^{ \mathbb{ A}^1}(X)(F)=\mathcal{ S}^2(X)(F)$. Furthermore, by combining this result with Morel's characterization of $\mathbb{ A}^1$-connected spaces in terms of the triviality of field-valued sections of $\pi_0^{ \mathbb{ A}^1}$, we provide a complete characterization of $\mathbb{ A}^1$-connected smooth quadratic hypersurfaces in $\mathbb{ A}^n_k$.
Categories: math.AG