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

TL;DR

This paper computes the sheaf of -connected components for certain threefolds obtained by blowing up varieties with -fibrations over surfaces, establishing -invariance in these cases.

Contribution

It explicitly determines the -connected components sheaf for blow-ups of threefolds over specific surfaces, extending understanding of -invariance in these geometries.

Findings

Sheaf of -connected components is explicitly determined.

-invariance of the sheaf is verified for these threefolds.

Results apply to blow-ups over -rigid or non-uniruled surfaces.

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.