A¹-connected components of schemes

A conjecture of Morel asserts that the sheaf of A^1-connected components of a simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that A^1-connected components of smooth k-schemes coincide with their A^1-chain-connected components and are birational invariants of smooth proper schemes. In this article, we exhibit examples of schemes for which Asok-Morel's conjectures fail to hold and whose Sing_* is not A^1-local. We also give equivalent conditions for Morel's conjecture to hold. A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field k.