Abstract

A conjecture of Morel asserts that the sheaf of $\mathbb A^{1$-connected components of a space is $\mathbb A}1$-invariant.  We will discuss how the sheaves of ``naive” as well as ``genuine” $\mathbb A^{1$-connected components of a smooth projective birationally ruled surface can be determined using purely algebro-geometric methods.  We will discuss a proof of Morel's conjecture for a smooth projective surface birationally ruled over a curve of genus > 0 over an algebraically closed field of characteristic 0.  If time permits, we will indicate why the naive and genuine $\mathbb A}1$-connected components of such a birationally ruled surface do not coincide if the surface is not a minimal model and discuss some open questions and specultions regarding the situation in higher dimensions.  The talk is based on joint work with Chetan Balwe.