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We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphismsfrom a smooth henselian local scheme into a smooth projective surface, which isbirationally ruled over a curve of genus $> 0$. We consequently determine thesheaf of naive $\mathbb A^1$-connected components of such a surface and showthat it does not agree with the sheaf of its genuine $\mathbb A^1$-connectedcomponents when the surface is not a minimal model. However, the sections ofthe sheaves of both naive and genuine $\mathbb A^1$-connected components overschemes of dimension $\leq 1$ agree. As a consequence, we show that theMorel-Voevodsky singular construction on a smooth projective surface, which isbirationally ruled over a curve of genus $> 0$, is not $\mathbb A^1$-local ifthe surface is not a minimal model.

Naive $\mathbb A^1$-homotopies on ruled surfaces