Affine Space Fibrations

Abstract

We discuss various aspects of affine space fibrations \(f : X \rightarrow Y\) including the generic fiber, singular fibers and the case with a unipotent group action on X. The generic fiber \(X_\eta \) is a form of \({\mathbb A}^n\) defined over the function field k(Y) of the base variety. Singular fibers in the case where X is a smooth (or normal) surface or a smooth threefold have been studied, but we do not know what they look like even in the case where X is a singular surface. The propagation of properties of a given smooth fiber to nearby fibers will be studied in the equivariant case of Abhyankar-Sathaye Conjecture in dimension three. We also treat the triviality of a form of \({\mathbb A}^n\) if it has a unipotent group action. Treated subjects are classified into the following four themes 1. Singular fibers of \({\mathbb A}^1\)- and \({\mathbb P}^1\)-fibrations, 2. Equivariant Abhyankar-Sathaye Conjecture in dimension three, 3. Forms of \({\mathbb A}^3\) with unipotent group actions, 4. Cancellation problem in dimension three.