A conjecture of Morel asserts that the sheaf π A 1 0 (X ) of A 1 -connected components of a simplicial sheaf X is A 1 -invariant. A conjecture of Asok and Morel asserts that the sheaves of A 1 -connected components of smooth schemes over a field coincide with the sheaves of their A 1 -chain-connected components. Another conjecture of Asok and Morel states that the sheaf of A 1 -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 * is not A 1 -local. We also give equivalent conditions for Morel's conjecture to hold and obtain an explicit conjectural description of π A 1 0 (X ). A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field.

Abstract. We prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.

We define an analog in characteristic p > 0 of the proalgebraic completion of the topological fundamental group of a complex manifold.

No abstract