A1A1-connected components of schemes

A conjecture of Morel asserts that the sheaf π0A1(X) of A1A1-connected components of a simplicial sheaf XX is A1A1-invariant. A conjecture of Asok and Morel asserts that the sheaves of A1A1-connected components of smooth schemes over a field coincide with the sheaves of their A1A1-chain-connected components. Another conjecture of Asok and Morel states that the sheaf of A1A1-connected components is a birational invariant of smooth proper schemes. In this article, we exhibit examples of schemes for which conjectures of Asok and Morel fail to hold and whose Sing⁎Sing⁎ is not A1A1-local. We also give equivalent conditions for Morel's conjecture to hold and obtain an explicit conjectural description of π0A1(X). A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field.