Naive A1-Homotopies on Ruled Surfaces

We explicitly describe the 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 A(1)-connected components of such a surface and show that it does not agree with the sheaf of its genuine A(1)-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine A(1)-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 A(1)-local if the surface is not a minimal model.