Families of Group Actions, Generic Isotriviality, and Linearization

TL;DR

This paper establishes a generic equivalence theorem for affine morphisms, proving local triviality of certain fibrations and demonstrating linearizability of reductive group actions on affine spaces.

Contribution

It introduces a generic equivalence theorem for affine morphisms and applies it to prove local triviality of affine fibrations and linearization of reductive group actions.

Findings

Affine morphisms with isomorphic fibers become isomorphic after an etale base change.

Affine $A^1$-fibrations over normal varieties are locally trivial in the Zariski topology.

Faithful reductive group actions on $A^3$ are linearizable.

Abstract

We prove a "Generic Equivalence Theorem which says that two affine morphisms $p: S \to Y$ and $q: T \to Y$ of varieties with isomorphic (closed) fibers become isomorphic under a dominant etale base change $\phi: U \to Y$. A special case is the following result. Call a morphism $\phi: X \to Y$ a "fibration with fiber $F$" if $\phi$ is flat and all fibers are (reduced and) isomorphic to $F$. Then an affine fibration with fiber $F$ admits an etale dominant morphism $\mu: U \to Y$ such that the pull-back is a trivial fiber bundle: $U\times_Y X \simeq U\times F$. As an application we give short proofs of the following two (known) results: (a) Every affine $\A^1$-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine $\A^2$-fibration over a smooth curve is locally trivial in the Zariski-topology. We also study families of reductive group actions on $\A^2$ parametrized by curves and show that every faithful action of a non-finite reductive group on $\AA^3$ is linearizable, i.e. $G$-isomorphic to a representation of $G$.