Naive 𝔸1-Homotopies on Ruled Surfaces

Abstract

We explicitly describe the A1-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 A1-connected components of such a surface and show that it does not agree with the sheaf of its genuine A1-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine A1-connected components over schemes of dimension ≤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 A1-local if the surface is not a minimal model.