A^1-connected components of ruled surfaces

TL;DR

This paper proves Morel's conjecture for all smooth projective surfaces over algebraically closed fields of characteristic zero by determining the sheaf of $A^1$-connected components of certain ruled surfaces using algebraic geometry.

Contribution

It establishes the $A^1$-invariance of the sheaf of $A^1$-connected components for a class of ruled surfaces, confirming Morel's conjecture in this case.

Findings

The sheaf of $A^1$-connected components is determined for certain ruled surfaces.

Morel's conjecture holds for all smooth projective surfaces over algebraically closed fields of characteristic zero.

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 algebro-geometric 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 $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.