${\mathbb{A}}^1$-connected components of blow-up of threefolds fibered over a surface

Abstract

Over a perfect field, we determine the sheaf of ${\mathbb{A}}^1$-connected components of a class of threefolds given by the Blow-up of a variety admitting a ${\mathbb{P}}^1$-fibration over either an ${\mathbb{A}}^1$-rigid or a non-uniruled surface, along a smooth curve. As a consequence, we verify that the sheaf of ${\mathbb{A}}^1$-connected components for such varieties is ${\mathbb{A}}^1$-invariant.

19 pages, major restructuring/changes in exposition after initial journal report

References in corpus (1)

• Naive ${\mathbb{A}}^1$-homotopies on ruled surfaces