A1–connected components of ruled surfaces

Abstract

A conjecture of Morel asserts that the sheaf of ${\mathbb{A}}^1$–connected components of a space is ${\mathbb{A}}^1$–invariant. Using purely algebrogeometric methods, we determine the sheaf of ${\mathbb{A}}^1$–connected components of a smooth projective surface, which is birationally ruled over a curve of genus . As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.