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

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 ${\mathbb{A}}^m$ $\to$ ${\mathbb{A}}^n$ with all fibers isomorphic to ${\mathbb{A}}^2$ is the trivial ${\mathbb{A}}^2$-bundle.

References in corpus (2)

• Venereau-type polynomials as potential counterexamples
• Simple birational extensions of the polynomial ring $\C^{[3]}$