A 1 – connected components of ruled surfaces

Abstract

A conjecture of Morel asserts that the sheaf of A1–connected components of a space is A1–invariant. Using purely algebrogeometric methods, we determine the sheaf of A1–connected components of a smooth projective surface, which is birationally ruled over a curve of genus >0. As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic 0.