$\mathbb A^1$-connected components of affine quadrics

TL;DR

This paper investigates the $A^1$-connected components of smooth quadratic hypersurfaces in affine space, showing the stabilization of naive $A^1$-connected components at the second iteration and characterizing $A^1$-connected quadrics.

Contribution

It demonstrates that the sheaf of $A^1$-connected components stabilizes after two iterations and provides a complete characterization of $A^1$-connected quadratic hypersurfaces.

Findings

Stabilization of $A^1$-connected components at the second iteration.

Complete characterization of $A^1$-connected quadratic hypersurfaces.

Connection with Morel's characterization of $A^1$-connected spaces.

Abstract

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 $\mathbb{A}^n_k$.