Families of Group Actions, Generic Isotriviality, and Linearization

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 $G$.