Naive 𝔸1-Homotopies on Ruled Surfaces

Abstract

Abstract We explicitly describe the $\mathbb{A}^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb{A}^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb{A}^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb{A}^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, we show that the Morel–Voevodsky singular construction on a smooth projective surface, which is birationally ruled over a curve of genus $> 0$, is not $\mathbb{A}^1$-local if the surface is not a minimal model.