# $\mathbb{A}^2$ -Fibrations between affine spaces are trivial $\mathbb{A}^2$-bundles

## TL;DR

The paper establishes a criterion for flat affine plane fibrations over smooth schemes to be trivial bundles, and confirms a version of the Dolgachev-Weisfeiler Conjecture for such fibrations with fibers isomorphic to .

## Contribution

It provides a criterion for triviality of -bundles and proves a conjecture for fibrations with all fibers isomorphic to .

## Key findings

- A flat fibration with affine plane fibers over a smooth scheme is a Zariski locally trivial -bundle.
- Any flat -fibration over  with all fibers isomorphic to  is trivial.
- The result applies to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations.

## Abstract

We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $\mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations: a flat fibration $\mathbb{A}^m$ $\rightarrow$ $\mathbb{A}^n$ with all fibers isomorphic to $\mathbb{A}^2$ is the trivial $\mathbb{A}^2$-bundle.